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!
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