Y. Yue J. Sun K. L. Gunter D. J. Michalek J. W. Sutherland Dept. of Mechanical Engineering— Engineering Mechanics, Michigan Technological University, Houghton, MI 49931 1 Character and Behavior of Mist Generated by Application of Cutting Fluid to a Rotating Cylindrical Workpiece, Part 1: Model Development Increasing attention is being devoted to the airborne emissions resulting from a variety of manufacturing processes because of health, safety, and environmental concerns. In this two-part paper, a model is presented for the amount of cutting fluid mist produced by the interaction of the fluid with the rotating cylindrical workpiece during a turning operation. This model is based on relationships that describe cutting fluid atomization, droplet settling, and droplet evaporation. Experiments are performed to validate the model. In Part 1 of the paper, the emphasis is on model development. In the model, thin film theory is used to determine the maximum fluid load that can be supported by a rotating cylindrical workpiece; rotating disk atomization theory is applied to the turning process to predict the mean size of the droplets generated by atomization; and expressions for both the evaporation and settling behavior are established. Droplet size distribution and mass concentration predictions are used to characterize the fluid mist. Model predictions indicate that the droplet mean diameter is affected by both fluid properties and operating conditions, with cutting speed having the most significant affect. Model predictions and experimental results show that the number distribution of droplets within the control volume is dominated by small droplets because of the settling and evaporation phenomena. In Part 2 of the paper, the cutting fluid mist behavior model is validated using the results obtained from a series of experiments. @DOI: 10.1115/1.1765150# Introduction Cutting fluids are widely used to cool and lubricate, flush away chips, and inhibit corrosion during machining operations such as drilling, turning, and grinding. However, significant negative effects, in terms of environmental, health, and safety consequences, are associated with the use of cutting fluids. In particular, the production of cutting fluid mist erodes air quality and has been linked to undesirable health effects @1#. Cutting fluid mist droplets could conceivably be produced by two mechanisms: atomization and vaporization/condensation ~depicted in Fig. 1!. Atomization can result from the interaction of the fluid with both stationary and rotating elements within the machine tool system. Vaporization may take place as small amounts of fluid are exposed to the hightemperature surfaces that result from the heat generated in the cutting zone. This vapor may subsequently condense around spontaneously generated liquid nuclei or other foreign particles to form mist droplets. The dominant mist formation mechanism will depend upon the machining process and the cutting fluid application strategy. Exposure to cutting fluid mist has been the subject of numerous studies @1–3#. In order to minimize worker exposure to fluid mists, common control strategies include enclosing the machine tool, using air filters or mist collectors, and adding antimisting agents to the fluid. These mist control methods generally represent an added cost to the process and may do little to prevent mist from forming. An alternative strategy is to modify the machining process itself to minimize the formation of cutting fluid mist. Such a strategy requires a mechanistic understanding of the effect of process conditions on mist formation. Contributed by the Manufacturing Engineering Division for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received March 10, 2003; revised Feb. 6, 2004. Associate Editor: Dong-Woo Cho. From an experimental perspective, some efforts have examined the influence of process conditions on mist formation. Gunter and Sutherland @4# performed a series of turning process experiments that examined the effects of spindle speed, workpiece diameter, nozzle diameter, proximal location, and oil concentration on cutting fluid mist mass concentration and droplet size distribution. Some efforts have focused on the development of a model to predict the mass concentration, mean droplet size, and the size distribution of droplets produced during a turning process @5,6#. However, a fully validated model that predicts all of the generated mist characteristics and the behavior of the mist post generation has yet to be presented. This two-part paper is focused on predicting the amount and character of cutting fluid mist produced by the interaction of the fluid with the rotating cylindrical workpiece during a turning operation. This part of the paper ~Part 1! is devoted to the development of a model for mist generation and behavior, and Part 2 emphasizes the experimental validation of the model. The model considers the formation of cutting fluid droplets by atomization, the settling of droplets, and the evaporation of droplets. The model is capable of predicting the mist mass concentration and size distribution, each of which has its own impact on determining the risk to worker health. OSHA standards are expressed in terms of the mass concentration of airborne particulate, and the droplet size distribution plays a key role in the deposition efficiency within the various regions of the human respiratory tract. Determining the mist size distribution is also important because: i! there is a strong relationship between droplet size and droplet settling behavior, and ii! the efficiency of mist collection/containment systems depends on the character of the droplet size distribution. It will be seen that the settling phenomenon is critical to the timevarying character of the mist size distribution and mass concentration. Journal of Manufacturing Science and Engineering Copyright © 2004 by ASME AUGUST 2004, Vol. 126 Õ 417 Fig. 1 Cutting fluid mist formation mechanisms in turning 2 Overview of Cutting Fluid Mist Model This paper presents an effort to explore cutting fluid mist formation and behavior during a wet turning operation. The relevant mechanisms associated with the generation and behavior of the mist during this machining process include: • Mist generation mechanisms: atomization and vaporization/ condensation. • Mist behavior mechanisms: settling, coagulation, diffusion, and evaporation. Due to the complexity of the actual machining process, development of the air quality model is being conducted in several stages. This procedure allows an assessment of the individual mechanisms on the air quality and facilitates the validation of the model. The present model includes the following three mechanisms from those listed above: • Mist generated by atomization resulting from the interaction of the cutting fluid and a rotating cylindrical workpiece, • Droplet settling, and • Droplet evaporation. Relationships will be presented for each of these individual mechanisms. Part 2 of the paper will focus on the validation of these relationships, both individually and collectively. The relationship between the components of the cutting fluid mist model, as well as the experimental methods to be used for model validation, is depicted in Fig. 2. The continual generation of droplets is described by a statistical distribution for droplet size that has a geometric mean and standard deviation of D g and s g , respectively. The geometric mean droplet size (D g ) is predicted based on atomization theory, and the geometric standard deviation ( s g ) is determined using experimental data. Generated droplets are placed within the volume, and the settling and evaporation behavior of the droplets is simulated. This enables the prediction of the size distribution of the droplets and the mass concentration of airborne particulate at any point in time. The predicted airborne particulate size distribution is characterized with a mean, D p , and a standard deviation, s p . Also illustrated in Fig. 2 are the air sampling measurements that are used to infer information about the airborne particulate within the control volume. This information is in the form of a mass concentration value or a frequency histogram of droplet diameters. This type of experimental data will ultimately be compared with the model predictions in an effort to assess the adequacy of the model. 418 Õ Vol. 126, AUGUST 2004 Fig. 2 Relationship of droplet generation and mist behavior model components 3 Model for Mist Generation In a machining environment fluid mist may be created as a result of i! an atomization mechanism, and/or ii! a vaporization/ condensation process. Cutting fluid vapor can be generated from two distinct sources, namely the machining process and the open fluid sources. For the latter source, based on results from previous studies @7,8#, it was determined that at ambient conditions, and even at temperatures above ambient, fluid losses are primarily due to the evaporation of water, not the oil concentrate, and evaporative losses of water to the air do not represent a health concern. However, at the elevated temperatures present during a machining process, the vaporization/condensation mechanism may be important and therefore should be considered. However, the modeling of this mist generation mechanism is beyond the scope of the current paper, and therefore only those droplets formed by the atomization mechanism are considered herein. Atomization is the process by which a liquid disintegrates into droplets due to the unstable growth of an initially small disturbance. In a turning operation cutting fluid atomization can result from several sources including; impingement with the machine tool, interaction with the rotating cylindrical workpiece, splashing at the point of application, and interference with swarf. While all of these mechanisms should be included in a complete cutting fluid atomization model, the model presented herein includes only the interaction of the fluid with the rotating cylindrical workpiece. This mechanism was the first to be modeled as it was felt to be the most dependent on machining conditions and would therefore reveal the most information about variations in air quality due to process conditions. The remaining atomization mechanisms will be considered in future stages of the model development. In an effort to model the atomization of cutting fluid due to its interaction with the rotating workpiece, Yue et al. @7,8# conducted some preliminary work that considered the application of fluid to a rotating disk. It was reported that after the fluid contacts the surface of the disk, it develops into a film, flows outwardly and separates from the disk at its edge. Depending on the fluid discharge velocity, disk diameter, and rotating speed, as well as the physical properties of the liquid, there are three different disintegration modes that may be present @9#: drop mode, ligament formation mode, and film formation mode. These are shown in Fig. 3. In the present study, a vertically oriented fluid stream impinges normal to the surface of a cylindrical workpiece of radius R, rotating at an angular velocity v. The turning process that was investigated in this study is pictured in Fig. 4. The figure shows the Transactions of the ASME Fig. 3 Three modes of atomization: „a… drop formation, „b… ligament formation, and „c… film formation orientation and location of the fluid stream relative to the workpiece surface. This configuration was considered in order to isolate the interaction of the fluid with the rotating cylindrical workpiece. It should be noted that the fluid nozzle is located in close proximity to the workpiece in order to minimize mist generation by splashing @10# and the tool is not present so as to eliminate atomization resulting from its interaction with the fluid. As mentioned previously, these mist generation mechanisms will be included in future stages of model development. As is evident, a film of fluid develops on the workpiece and mist droplets are produced at two positions along the axis of the workpiece, henceforth referred to as rims. The formation of droplets at the rims can be likened to the rotating disk atomization process. Using this analogy, and thereby applying rotating disk atomization theory, the mean droplet diameter can be obtained. While it is convenient to discuss a mean drop size, an atomization process produces a wide range of drop sizes. This is due to the random nature of the atomization process, resulting from one or more of the following: i! numerous small satellite droplets are produced in addition to the large droplets, ii! multiple modes of disintegration can simultaneously exist within an atomization process, iii! cutting fluids are heterogeneous mixtures ~rather than homogeneous solutions! with varying physical properties, and iv! small oil droplets may be stripped from the emulsion rather than being produced by a classical atomization process @11#. The mist generation portion of the model uses a lognormal distribution, which is characterized by a geometric mean, D g , and a geometric standard deviation, s g , henceforth referred to simply as the mean and standard deviation, respectively, to describe the distribution of droplets that are generated. The determination of these quantities, as well as the corresponding mass concentration, will be discussed in the following sections. At this point it should be noted that the equations used to model the atomization process are expressed in Fig. 5 Two-dimensional schematic for fluid impingement on rotating cylinder and resulting flow and film formation terms of the rotational rate of the cylinder, while cutting speed is typically used in manufacturing applications. For clarity, the rotational rate will be used during the discussion of the model while both rotational rate and cutting speed will be presented in the model predictions. 3.1 Maximum Fluid Flow Rate. As previously discussed, as the vertically oriented cutting fluid stream impacts the rotating cylindrical workpiece a film of fluid develops on the workpiece. The amount of fluid that can be supported by the rotating cylinder is dependent on fluid properties, system geometry, and process conditions. For instance, for very large fluid application flow rates, some of the fluid will follow the pattern illustrated in Fig. 5, while much of it will simply drain off the cylinder due to gravitational effects. As the flow rate is reduced and the spindle speed is held constant, eventually the drainage will stop and all the fluid flow will contribute to rim formation. Using the given system geometry, fluid properties, and process conditions, a maximum fluid flow rate, q max5LQ v R 2 , (1) can be determined, once Q, which is a function of the width, L, and the thickness, h, of the liquid film are calculated. This is then compared to the applied fluid flow rate to determine the mass concentration of airborne particulate generated by atomization. The maximum fluid flux can be determined from the maximum film thickness and the width of the fluid film that forms on the rotating cylinder. Thin film theory @12# can be applied to determine the film thickness. Preziosi and Joseph @13# describe this situation using a non-dimensional Navier-Stokes equation in standard polar coordinates ~r,u!, together with the assumption that the radial component of the fluid velocity can be neglected. A no-slip condition is used on the cylinder surface and no shear stress is present on the free surface. Kelmanson @12# used thin film theory to derive an expression for the nondimensional fluid film thickness, h, in terms of the nondimensional fluid flux across the fluid film, Q, and the angle, u. Neglecting higher order terms, the power series for h is: h ~ Q, u ! 5Q2 F G 1 2 g cos u 1 3 Q 1 1 Q . 2 3 2 (2) From Eq. ~2!, an existence condition for steady state flow may be developed: C5 Fig. 4 Turning operation with fluid stream application Journal of Manufacturing Science and Engineering 288g 4 Q 2 124g 2 ~ 5Q13 ! 21 64g 3 1 ~ 1116g 2 ~ 113Q !! 3/2 <1, (3) where, C51 represents the limiting condition for steady flow. The value of Q associated with C51 ~for a given Stokes number! represents the maximum non-dimensional fluid flux and can be estimated by numerically solving Eq. ~3!. Thus, Eq. ~2! may be evaluated and after multiplying by R, the actual film thickness is obtained. AUGUST 2004, Vol. 126 Õ 419 with the machining conditions, experimental observations indicate that the ligament mode is the dominate droplet formation mechanism. As such, it will be the focus of this paper. As previously mentioned, two rims are formed on the workpiece as depicted in Fig. 6. The fluid film in these two rims takes on a wave shape as a result of disturbances, and unstable waves lead to ligament formation. Disintegration of the ligaments at the circumference of the rim is similar to the disintegration of ligaments on a rotating disk, as depicted in Fig. 3~b!. Using the rotating disk relation for the mean droplet diameter @15#, the following expression is obtained: D̄51.23R 1/7 S DS DS D 1 Nl 2/7 r q l2 4zR3 z 2/7 r R 3v , (6) where q l is the total volumetric flow rate to form the ligaments. The number of ligaments, N l , is determined by an analysis of the disturbance of the liquid film at the circumference of the disk. Assuming that the number of ligaments is associated with the maximum growth rate of disturbance, N l can be calculated from the following equation: Fig. 6 Rims and ligaments in turning F AS H We5N l2 31 ~ 8N l 23 ! St 11 3.1.2 Fluid Film Width. The other quantity needed to determine q max is the width of the fluid film, L. When a vertically oriented fluid stream of radius r j and velocity U j impinges on the top of the rotating cylinder, the high pressure generated in the stagnation region causes the fluid to spread laterally along the workpiece axis, which produces a thin film as show in Fig. 5. The axial position at which the flow stops and rims form depends on the workpiece rotating speed, stream velocity, and the fluid properties. If the point of application of the fluid corresponds to an axial position of z50, the rims are associated with axial positions of z56L/2, as shown in Fig. 6. The actual fluid film width is related to the non-dimensional fluid film width by the expression, L52z s R. (4) The nondimensional fluid film width (2zs) can be estimated from the following equation @14#: 2z s '3.32Fr 0.23« 0.36, Fr5U 2j /(gR) (5) 1/2 is the Froude number, «51/2Re (r j /R) 2 , where and Re5U jr j /n is the Reynolds number of the fluid stream. Using q max , which is calculated from Eq. ~1!, the rate of change in the mass concentration of the airborne particulate can be determined assuming that all of the fluid that forms the film is atomized. For instance, if the applied fluid flow rate, q, is larger than q max for the given conditions, then some fluid will drain from the workpiece and the amount of the fluid that forms the ligaments (q l ) will equal q max . On the other hand, if the fluid flow rate, q, is smaller than q max , the fluid flow rate to form the ligaments will be q l 5q. ~For such a case, the film thickness should be calculated based on the actual fluid flow rate, q.! 3.2 Mean Droplet Diameter. The mass concentration of airborne particulate produced by the atomization process is one characteristic used to evaluate the air quality. As previously discussed, the other is the droplet size distribution, which is characterized by a mean droplet diameter and a standard deviation. For a turning process in which a vertically oriented fluid stream impinges normal to the workpiece surface, the cylindrical fluid film that is formed is always unstable regardless of the value of the Reynolds number based on the stream properties. It is clearly seen in Fig. 4 that rims develop about the rotating cylinder at two axial positions. For each rim, there are three possible liquid film disintegration modes, namely drop mode, ligament formation mode, and film formation mode. While the breakup mode will change 420 Õ Vol. 126, AUGUST 2004 11 1 N l St D GJ , (7) where We is the Weber number and St is the stability number and can be obtained by: St5 m2 . rs R (8) The mean droplet diameter obtained from Eq. ~6! will serve as the droplet geometric mean, D g , for the droplet generation distribution described below. 3.3 Droplet Size Distribution. The random character of cutting fluid mist generation leads to the production of droplets of varying sizes. The frequency with which droplets of different sizes are produced can be described by mathematical expressions, known as probability density functions or frequency distributions, whose parameters can be estimated from measured data. There are several theoretical frequency distributions for characterizing droplet size distributions @16#, including: lognormal, Rosin-Rammler, and power-law. As no single distribution function can represent the droplet size data in all cases, usually several theoretical distribution functions are compared to a given set of experimental data, and the function that best describes the data is selected @17#. The lognormal distribution has been used in this paper to depict the size distribution of droplets generated by atomization ~it will also be used to represent the size distribution of airborne mist!. The lognormal distribution was selected because the histograms of particle sizes have lognormal appearances. The lognormal probability density function associated with the particle count distribution for droplet generation is f ~ D !5 1 D ~ ln s g ! A2 p H F exp 2 1 ln~ D/D g ! 2 ln s g 2 GJ , (9) where the parameters D g and s g are the droplet geometric mean and standard deviation, respectively ~in the case of the airborne particulate size distribution, the parameters are D p and s p ). The spread in the distribution is characterized by the parameter s g : a small value of s g indicates a nearly monodisperse, narrow size distribution, and a large value of s g indicates a polydisperse, broad size distribution. Experimentally obtained data on particle sizes is often expressed using frequency histograms, with the fraction of the total particles collected displayed as a function of diameter. In terms of the theoretical density function, for the kth size range with an associated diameter D k , the fraction of the total droplets that lie Transactions of the ASME within a range of DD k about this diameter is obtained by integrating the lognormal density function across the range. This probability may be approximated as E Du f ~ D ! dD' f ~ D k ! DD k , 4.2 Model for Droplet Settling. At a location H below the top of the enclosure the rate of change in the number concentration associated with droplet settling for the kth size class can be expressed as: (10) S D dn k dt Dl where, D u 5DD k /21 A(DD k ) 2 14D 2k /2 and D l 5D u 2DD k . 4 Model for Mist Behavior The previous section focused on describing the distribution of droplets that are generated by atomization resulting from the impingement of a cutting fluid stream on a rotating cylindrical workpiece. The size distribution of the airborne particulate changes with time as a result of several processes: continuing droplet generation, diffusion, coagulation, settling, and evaporation. Following their production by the atomization process, the droplets are assumed to be suspended uniformly within an enclosure of volume V and height H. Within the enclosure, the number concentration, n k , for the kth size class is given by the expression: f ~ D k ! DD k n k 5N p , V (11) where N p is the total number of droplets within the enclosure. At any point within the enclosure, the rate of change in the number concentration for the kth size class can be described by a general dynamic equation with droplet generation, settling and evaporation effects, viz., S D S D S D dn k dn k 5 dt dt dn k dt 1 gen 1 set dn k dt . (12) evap Building upon the findings associated with number concentration, the dynamic equation ~for the kth size class! for mist mass concentration can be expressed as: S D S D S D dm k dm k 5 dt dt dm k dt 1 gen 1 set dm k dt . (13) evap The total mass concentration may then be obtained by adding the concentrations for each of the size classes: M5 (m k k . (14) Using the relations for mass concentration developed above, the dynamic behavior of the mass concentration can be simulated. 4.1 Model for Droplet Generation. The rate of change in the number concentration associated with droplet generation for the kth size class depends on the distribution shape and the rate at which droplets are generated. This may be expressed as: S D S D dn k dt 5 gen 6q l p D m3 DD k f g ~ D k ! , V S D S D S D dm k dt 5 gen dn k dt gen (16) (17) Journal of Manufacturing Science and Engineering set vk n k 0 H , (18) where the number concentration of particles in the kth class at time t and for an initial condition are, n k and n k 0 , respectively, and vk 5 r gD 2k /18m g is the settling velocity of droplets of size D k ~where m g is the viscosity of air!. Similarly, the rate of change in the mass concentration associated with settling can be expressed as: S D dm k dt 52 set v km k0 H 5 pr D 3k , 6 S DS D dn k dt set (19) where, m k 0 is the initial mass concentration of droplets in size range k. 4.3 Model for Droplet Evaporation. Previous sections have established relationships for droplet generation and mist settling behavior. At each time step, the relationships developed for generation and settling compute the rate of change in the number of droplets within a given droplet size class. Attention now shifts to the mathematical description of droplet evaporation, which like generation and settling is also expected to significantly affect the mist concentration and size distribution. It is reasonable to assume that a relationship for evaporation could be implemented similarly to that for settling in order to compute the droplet diameter changes at each time step. However, Kukkonen et al. @18# reported that significant evaporation will not occur for number concentrations greater than 13107 /m3 , and for the situation under consideration mist concentrations are usually well above this critical level. Indeed, the droplet number concentration increases rapidly once the cutting fluid application starts, and in a very short period of time the critical concentration level stated above is exceeded. However, once fluid application stops, the concentration decays exponentially at a rate of decrease that is largely associated with the settling mechanism. When the concentration drops below the critical level, evaporation may occur. An evaporation model has been established for situations in which the number concentration level is below the critical level. The evaporation model does not add or remove droplets from the control volume, but rather shifts droplets from one size class to a smaller size class. To characterize the evaporation effect for droplets in the kth size class, the non-equilibrium Langmuir-Knudsen evaporation law is applied, and droplet temperature non-uniformity is considered @19#. In order to account for the droplet temperature nonuniformity, generic Lagrangian equations are employed to describe the transient velocity, temperature, and mass of a droplet. The transient velocity ~which affects the Reynolds number! is determined by: (15) where, DD k is the size range, D m is the average mass diameter, f g is the density function for the distribution of generated droplets ~a lognormal distribution with a mean of D g , given by Eq. ~6!, and a standard deviation, s g , to be estimated from experiments!, and q l is the fluid flow rate to the ligaments that form droplets. For the kth size range the rate of change in the mass concentration due to droplet generation is: pr D 3k . 6 52 vk 5 r gD 2k C , 18m G c (20) where C c 5116.631028 /D k @ 2.51410.8 exp(20.55D k /6.6 31028 ) # is the Cunningham correction factor. The transient mass ~which determines droplet size! is expressed as: S D dm k Sh m k 52 HM . dt 3Sc G t k (21) Finally, the transient temperature ~which affects the rate of heat transfer between the droplet and the surrounding gas! is given by: AUGUST 2004, Vol. 126 Õ 421 Fig. 7 Maximum cutting fluid flow rate capable of being sustained by rotating workpiece S D S D dT k L v ṁ k Nu u 1 5 . f 2 ~ T G 2T k ! 1 dt 3 Pr G t k Cl mk (22) where ṁ k 5dm k /dt is negative for evaporation. The subscripts on the variables denote droplets in the kth size class ~k!, the gas phase away from droplet surface ~G!, the vapor phase of the evaporate ( v ), and the liquid phase ~l!. The particle time constant for Stokes flow is t k 5 r D 2k /18m G , and f 2 5 b /(e b 21) is an evaporative heat transfer correction factor, and the Nusselt and Sherwood numbers are empirically modified for convective corrections to heat and mass transfer. T G is the gas temperature, L v is the latent heat of evaporation, and the ratio of the gas ~constant pressure! heat capacity to that of the liquid phase is u 1 5C P G /C l . Finally, H M represents the specific driving potential for mass transfer. The diameter change of a droplet due to evaporation can be obtained from Eq. ~20!. The equation is applied to each size class, and the number of droplets in the original size class is associated with a new size class. 5 Fig. 8 Effect of spindle speed on mean droplet diameter for several workpiece diameter and fluid flow rate combinations excess of the maximum produce fluid runoff, or for a fixed fluid application rate, rotational rates below the cutoff will result in runoff. The figure also illustrates that higher cutting speeds can support the application of larger fluid flow rates without runoff. Further experimental investigation indicates that cutting speed and nozzle diameter are the two primary parameters that affect the maximum flow rate. This will be discussed further in Part 2 of this paper. Predictions of droplet mean diameter behavior as a function of spindle speed for different combinations of workpiece diameter and cutting fluid application flow rate are presented in Fig. 8. These results were generated using Eq. ~6! with a nozzle diameter of 6.35 mm. From the figure it is clear that workpieces with larger diameters and higher spindle speeds produce smaller droplets. Also apparent is that an increase in the fluid application flow rate results in increased droplet sizes. Displaying these same results versus cutting speed leads to slightly different conclusions. As can be seen from Fig. 9, the cutting speed is the primary factor effecting the predicted mean droplet size. Workpiece diameter and flow rate lead to only slight changes in mean droplet size. Model Predictions The model describing the droplet generation and mist behavior developed above can be used to analytically assess the variation in air quality for a variety of machining process conditions. In this section various model predictions will be presented so that trends can be identified. This will aid in the validation of the model that is presented in Part 2 of the paper. For all simulations, unless otherwise stated, the fluid used is 10% soluble oil, for which the thermophysical properties have been previously established as @20,21#: r5981 kg/m3, z50.03 N/m, and m l 51061 31026 kg/~m•s). One of the important features of the atomization mechanism associated with droplet generation is the amount of fluid that adheres to the rotating workpiece and contributes to ligament formation. For a given nozzle diameter (D n 56.35 mm) and workpiece diameter (D w 5104.8 mm), the predicted maximum fluid flow rate that a rotating workpiece is capable of sustaining for a given rotational rate is determined by Eq. ~5!. The values of q max are plotted as a function of both spindle speed and cutting speed in Fig. 7. In addition to indicating the maximum flow rate that can be supported by the cylinder at a given rotational rate, Fig. 7 also indicates regions of runoff and no runoff. The figure shows that for a given cutting speed ~or spindle speed!, fluid flow rates in 422 Õ Vol. 126, AUGUST 2004 Fig. 9 Effect of cutting speed on mean droplet diameter for several workpiece diameter and fluid flow rate combinations Transactions of the ASME Table 1 Cutting fluid properties Cutting Fluid water 10% soluble oil 10% synthetic fluid Density ~r! kg/m3 Viscosity ( m l ) kg/~m•s! Surface Tension ~z! N/m 997 981 994 790 3 1026 1061 3 1026 904 3 1026 0.074 0.030 0.026 The effect of cutting fluid type, or composition, on the atomization mechanism can also be explored using the model. The properties for the three different cutting fluids @20,21# under consideration are listed in Table 1. Using Eq. ~6! with a nozzle diameter of 6.35 mm, workpiece diameter of 104.8 mm, and the listed fluid properties, the predicted mean droplet diameter as a function of spindle speed and cutting speed are determined. The results are presented in Fig. 10. The figure reveals that the fluid type can significantly impact the mean droplet diameter. The results shown in Figs. 7–10 illustrate the effect of several variables and indicate that cutting speed is an important parameter that affects the generation of droplets produced by the atomization process. Predictions of the mist behavior ~dynamic changes in the mass concentration and droplet size distribution! within the enclosure may also be obtained. These predictions require knowledge of the standard deviation, s g , associated with the distribution of generated droplets. In Part 2 of this paper, experimental data will be presented to estimate s g , however, to illustrate the predictive capabilities of the model, a value for s g of 4.0 is presently assumed. Again 10% soluble oil is used at a flow rate of 3.4 L/min, with D n 56.35 mm, V51.02 m3 , H50.95 m, and a spindle speed is 2000 rpm. It should be noted that for this spindle speed and flow rate combination no runoff occurs ~as indicated in Fig. 7!, and therefore all of the applied cutting fluid is atomized. The first complete simulation conducted with the entire model predicts the variation in mass concentration while fluid is being applied to the rotating workpiece. Two workpiece sizes, 104.8 mm and 63.5 mm, with corresponding cutting speeds of 658.5 m/min and 399.0 m/min, respectively, were considered. The predicted geometric mean diameter, calculated using Eq. ~6!, is 312 mm for the large diameter workpiece and 477 mm for the smaller diameter workpiece. The predictions of PM10 ~particles having less than a Fig. 10 Effect of cutting speed and spindle speed on mean droplet diameter for several cutting fluid compositions Journal of Manufacturing Science and Engineering Fig. 11 Dynamic PM10 mass concentration behavior during fluid application 10 mm aerodynamic diameter! mass concentration, obtained using Eq. ~14!, for 4 minutes of fluid application are shown in Fig. 11. It can be seen that a larger PM10 mass concentration is associated with a smaller mean diameter ~larger workpiece diameter!. This behavior is as expected because the smaller mean diameter indicates that a greater number of droplets will be produced in the PM10 range. In addition, smaller droplets remain in the air longer due to a slower settling time, and therefore mass will accumulate at a faster rate. The next step in the simulation is to halt the cutting fluid application and allow the generated droplets to settle and evaporate in a quiescent environment. The predicted droplet size distribution for 15 and 35 minutes after the 4 minutes of fluid application are shown in Fig. 12. These results are obtained using Eq. ~12! to simulate the mist behavior. It is clearly seen from these results that the larger droplets produced by the atomization process are no longer present. This is due to both the settling and evaporation mechanisms. Fig. 12 Droplet distributions: 15 and 35 minutes after fluid application ends AUGUST 2004, Vol. 126 Õ 423 Summary The phenomena associated with cutting fluid mist formation by an atomization mechanism and subsequent behavior under the action of settling and evaporation have been explored in this first part of a two-part paper. Research from the technical literature associated with fluid impingement on a rotating disk has been described. A model is then presented for droplet generation by atomization resulting from the fluid interaction with a rotating cylindrical workpiece. This model has the following features: • The maximum fluid load that the rotating cylindrical workpiece can support is determined based on thin film theory. While these relations have been established previously by Kelmanson @12#, this is their first use in a cutting fluid application. Fluid flow rates exceeding the maximum will produce runoff, while lower flow rates lead to a mist formation rate equal to the fluid application flow rate. • Experimental observations suggest that the fluid rims and ligaments formed about the rotating cylinder during a turning process are analogous to the behavior of the fluid spun-off from a rotating disk. To predict mean droplet size, rotating disk atomization theory from the technical literature has been adapted to a rotating cylinder such as is used in a turning process. • As suggested by Hinds @16#, a lognormal distribution has been used to characterize the stochastic nature of the atomization process. • A novel approach to obtain the droplet distribution has been established. The droplet distribution within a given control volume is described as a function of the droplets entering the volume ~generated through atomization!, evaporating, and leaving the volume ~settling!. Expressions are derived for each behavior. • Mass concentration is predicted. Again incorporating the effect of continuing droplet generation, evaporation, and settling. • Predictions have been made for both the droplet distribution and mass concentration under different machining conditions. The behavior of the model predictions seems promising and consistent with expected trends. The second part of this paper will focus on validating the model. Nomenclature C c 5 Cunningham correction factor C l 5 heat capacity of liquid ~J/kg•K! C P G 5 heat capacity of ambient gas ~J/kg•K! D̄ 5 mean droplet diameter for rotating disk atomization ~mm! D g 5 geometric mean droplet diameter ~mm! D k 5 midpoint diameter of droplets in kth size class ~mm! D l 5 lower limit on range of droplets within DD k ( m m) D m 5 droplet diameter of average mass ~mm! D n 5 jet or nozzle diameter ~mm! D p 5 predicted geometric mean droplet diameter within the mist zone ~mm! D u 5 upper limit on range of droplets within DD k ( m m) D w 5 workpiece diameter ~mm! f 1 5 Stokes drag correction f 2 5 evaporative heat transfer correction factor Fr 5 Froude number, Fr5U 2j /(gR) f (D) 5 density function associated with the droplet size distribution f (D k ) 5 probability density evaluated at diameter D k f g (D k ) 5 distribution of generated droplets f m (D k ) 5 density function of measured droplet size data ~fraction/mm! g 5 acceleration due to gravity ~m/s2! H 5 height of mist zone ~enclosure! ~m! H M 5 specific driving potential for mass transfer h 5 nondimensional fluid film thickness L 5 actual fluid film width ~m! 424 Õ Vol. 126, AUGUST 2004 L v 5 latent heat of evaporation ~J/kg! M 5 total mass concentration ~mg/m3! m k 5 mass concentration of droplets in kth size class ~mg/m3! N 5 spindle speed ~rpm! N k 5 total number of droplets within kth size class N l 5 ligament number N p 5 total number of droplets within the mist zone Nu 5 Nusselt number n k 5 number concentration of droplets in kth size class ~1/m3! Pr G 5 Prandtl number of surrounding gas Q 5 nondimensional fluid flux across the fluid film q 5 cutting fluid stream volumetric flow rate ~m3/s! q l 5 total volumetric flow rate to form the ligaments ~m3/s! q max 5 maximum fluid flow rate associated with the maximum film thickness attainable for a given spindle speed ~m3/s! R 5 workpiece radius ~m! Re 5 Reynolds number r, u, z 5 cylindrical coordinates—radial, angular, and axial directions, respectively r j 5 cutting fluid stream radius ~m! Sc G 5 Schmidt number of surrounding gas Sh 5 Sherwood number St 5 stability number t 5 time ~s! T G 5 temperature of surrounding gas ~K! T k 5 temperature of droplets in kth size class ~K! U j 5 cutting fluid stream velocity ~m/s! u, v , w 5 tangential, axial, and radial fluid flow velocity components, respectively ~m/s! u d , v d 5 horizontal and vertical single droplet velocity components, respectively ~m/s! V 5 volume of the mist zone ~enclosure! ~m3! vc 5 cutting speed, v c 5(N• p •D w )/1000 (m/min) vk 5 settling velocity of droplets of size D k (m/s) We 5 Weber number, We5 rv 2 R 3 / z z s 5 half width of fluid film in non-dimensional expression g 5 Stokes number, g 5 r l gR/ v m l DD k 5 interval of droplet size ~mm! z 5 cutting fluid surface tension ~N/m! h 5 small radial position within fluid film, 0< h <h( u ) m g 5 dynamic viscosity of air ~kg/m•s! m l 5 dynamic viscosity of the cutting fluid ~kg/m•s! n 5 kinematic viscosity of the cutting fluid ~m2/s! r 5 cutting fluid density ~kg/m3! s g 5 geometric standard deviation of droplet size distribution s p 5 predicted geometric standard deviation of droplet size distribution after settling and evaporation t d 5 particle time constant for Stokes flow t r u 5 shear stress on free surface of fluid film C 5 limiting condition for steady flow v 5 workpiece angular velocity ~rad/s! References @1# Hands, D., Sheehan, M. 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