Numerical Thermodynamic Analysis of Two-Phase Solid

Hindawi Publishing Corporation
Advances in Mechanical Engineering
Volume 2014, Article ID 921831, 17 pages
http://dx.doi.org/10.1155/2014/921831
Research Article
Numerical Thermodynamic Analysis of Two-Phase Solid-Liquid
Abrasive Flow Polishing in U-Type Tube
Junye Li,1,2 Zhaojun Yang,1 Weina Liu,2 and Zemin Qiao2
1
2
College of Mechanical Science and Engineering, Jilin University, Changchun 130022, China
College of Mechanical and Electric Engineering, Changchun University of Science and Technology, Changchun 130022, China
Correspondence should be addressed to Zhaojun Yang; [email protected]
Received 18 April 2014; Revised 8 June 2014; Accepted 17 June 2014; Published 27 August 2014
Academic Editor: Weihua Cai
Copyright © 2014 Junye Li et al. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
U-type tubes are widely used in military and civilian fields and the quality of the internal surface of their channel often determines
the merits and performance of a machine in which they are incorporated. Abrasive flow polishing is an effective method for
improving the channel surface quality of a U-type tube. Using the results of a numerical analysis of the thermodynamic energy
balance equation of a two-phase solid-liquid flow, we carried out numerical simulations of the heat transfer and surface processing
characteristics of a two-phase solid-liquid abrasive flow polishing of a U-type tube. The distribution cloud of the changes in the inlet
turbulent kinetic energy, turbulence intensity, turbulent viscosity, and dynamic pressure near the wall of the tube were obtained.
The relationships between the temperature and the turbulent kinetic energy, between the turbulent kinetic energy and the velocity,
and between the temperature and the processing velocity were also determined to develop a theoretical basis for controlling the
quality of abrasive flow polishing.
1. Introduction
Abrasive flow machining is a polishing method by which
the pressure applied by the back-and-forth flow of a soft
abrasive viscoelastic medium is used to polish the surface of
the machined part [1]. The hard and sharp edges of the grains
in the abrasive flow are used as cutting tools for achieving a
certain degree of polishing. The advantage of abrasive flow
machining is its applicability to channels that cannot be easily
accessed by general tools [2, 3]. The even and progressive
abrasion of the surface and corners of the worked parts by
the grains can be used to accomplish deburring, polishing,
and chamfering. It has been experimentally confirmed that
abrasive flow machining can be used to significantly improve
the surface quality of nonlinear runners [4]. The machining
mechanism is illustrated in Figure 1.
Several studies on abrasive flow machining have been
conducted and reported. Jain et al. [5–9] experimentally
investigated the effect of the central force on the process
and obtained the relationship among the shaft velocity, cycle
index, and grain size, with the purpose of improving the
surface roughness of the grain and the material removal.
They also investigated the modeling of the abrasive media
by modifying the standard Maxwell model of elastomers to
the generalized Maxwell model. With the assumption that
the material removal by shear stressing was dependent on the
bonding of the abrasive grains, a fundamental material model
that could be easily integrated in conventional simulation
programs was developed and presented. Sankar et al. [10–12]
also discussed the effect of rotating abrasive flow machining
on the surface morphology of the workpiece and analyzed the
relationship between the rotational velocity of the workpiece
and the effective abrasion distance in the polishing area using
the theoretical and experimental spiral path lengths. They
also obtained the relationship among the rotational velocity
of the workpiece, the rate of change of the surface roughness,
and the amount of removed material. Furumoto et al. [13]
studied the interior of the injection mold used for free
abrasive finishing machining and found that high-velocity
free-flowing abrasion increases the kinetic energy of the
2
Advances in Mechanical Engineering
Cylinder
Cylinder
Piston
Piston
Streamer
(medium)
Clamped
DIE
Clamped
DIE
Streamer
(medium)
Figure 1: Principle of abrasive flow machining.
(a)
(b)
Figure 2: U-type tube model.
grains, which in turn increases the possibilities of contact
between the grains and the inner surface, thereby improving
the polishing quality. Kar et al. [14] put forward a method for
measuring the axial force of single abrasive flow polishing of
AISI4140, a material used for automobile hydraulic cylinders.
They investigated the effect of the medium pressure and
processing time on the surface quality of the workpiece. Ji
et al. [15, 16] used the Preston equation and its modified
coefficient of VOF to model the softness of the structural
surface of an abrasive flow. By numerical simulation, they
found that the average velocity of the abrasive particles in the
tube line increased with increasing inlet flow rate. Using the
cutting experience formula for soft abrasive flow, the Euler
multiphase flow model, and the realizable k-𝜀 turbulence
model of a V-type textural semiannular liquid-solid flow with
different particle concentrations, they performed numerical
simulations to investigate the turbulent velocity for different
types of particles. By examining the wall pressure, they
determined the optimum particle size for soft abrasive flow
polishing and also found that the shape and structure of the
microchannel significantly affected the process. Wan et al.
[17] proposed a simple zero-order semimechanistic approach
to the analysis of two-way abrasive flow machining. This
was prompted by the need to reduce the number of timeconsuming and labor-intensive experimental trials.
Wang et al. [18] also used a numerical method to design
effective passageways for producing a smooth surface in a
complex hole by abrasive flow machining. They examined
the current method and found that the shear forces of the
polishing process and the flow properties of the machining
medium determined the smoothness of the entire surface.
In the present study, the existing theory was examined
and simulation experiments were performed under specific
conditions. The study was prompted by the fact that unless
the grinding viscosity temperature characteristics of the
particle flow of abrasive flow polishing are determined and
the relevant quality relationships are established, the existing
theory would remain incomplete and there would be limited
Advances in Mechanical Engineering
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Figure 3: Mesh of U-type tube.
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Figure 4: Turbulent kinetic energy cloud for different inlet conditions of U-type tube channel.
universal guidelines for the process. In this study, the effects
of the viscosity temperature characteristics of abrasive flow
polishing on the quality of the produced surface were investigated with the purpose of developing the relationships among
the process temperature, turbulent kinetic energy, turbulence
intensity, velocity, and dynamic pressure. It is expected that
such would be used to improve the polishing quality of the
process and further develop its theory.
2. Numerical Thermodynamic Model of
Two-Phase Solid-Liquid Flow
In the mass conservation and momentum conservation
equations, the pressure and velocity are unknown physical
parameters. In a two-phase solid-liquid flow, the viscosity and
density often change, and the dynamic viscosity of the fluid
medium can be determined when large temperature changes
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4.29e + 02
3.60e + 02
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(c)
Figure 5: Turbulence intensity images for different inlet conditions of U-type tube channel.
occur. The density of a liquid is significantly related to its
compressibility, and variations in the density can be ignored
for small compressibilities. Hence, the energy conservation
equation can be used to determine the temperature at each
point in a flow field based on the other variables, which
include the pressure and velocity.
2.1. Energy Conservation Equation. The energy conservation
law is a basic law that a flow through a heat exchange system
is required to satisfy. The law states that the energy of an
element in a fluid flow is equal to the net increase in energy of
the element plus the physical work done on it by the surface
forces:
𝜕 (𝜌𝐸)
+ ∇ ⋅ [V (𝜌𝐸 + 𝑃)]
𝜕𝑡
(1)
= ∇ ⋅ [𝑘eff ∇𝑇 − ∑ ℎ𝑞 𝐽𝑞 + (𝜏eff ⋅ V)] + 𝑆ℎ ,
𝑞
where 𝐸 represents the kinetic energy and internal energy of
the fluid element: 𝐸 = ℎ − (𝑃/𝜌) + (V2 /2). In an abrasive flow,
the compressibility is small; hence, the enthalpy is given by
ℎ = ∑𝑞 𝑌𝑞 ℎ𝑞 + (𝑃/𝜌), where 𝑌𝑞 is the mass fraction of the 𝑞
phase. The specific enthalpy of the 𝑞 phase can be expressed
𝑇
as ℎ𝑞 = ∫𝑇 𝑐𝑝,𝑞 𝑑𝑇, where 𝐶𝑝,𝑞 is the specific heat capacity
ref
of the 𝑞 phase, 𝑇ref = 298.15 K, 𝑘eff is the effective thermal
conductivity of the fluid, and 𝐽𝑞 is the diffusion flux of the 𝑞
phase. The first three terms on the right-hand side of (1) are,
respectively, the thermal, diffusion, and viscous dissipation
terms of the energy transfer, and 𝑆ℎ may be the internal heat
of the source fluid, the exothermic or endothermic heat of a
chemical reaction, or some other custom term.
2.2. Particle Temperature Equation. The development of science and technology over recent years has led to the use of
the dynamic theory to determine the viscosity coefficient,
thermal conduction coefficient, particle phase pressure, and
other parameters. This has resulted in the development of the
particle temperature equation. In abrasive flow machining,
the collision between the abrasive particles and the wall
surface due to the irregular motion of the particles is similar
to the thermal motion of fluid molecules. The abrasive
Advances in Mechanical Engineering
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(a)
(b)
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(c)
Figure 6: Turbulent viscosity cloud for different inlet conditions of U-type tube channel.
particles in a turbulent two-phase flow continuously gain
energy, but their inelastic collisions with the wall also results
in constant dissipation of energy. The particle temperature
equation can therefore be suitably applied to the energy
variation of the particles of a two-phase liquid-solid flow:
In (2), 𝑃𝑠 and 𝜏𝑠 are, respectively, the compressive and
shear stresses of the particulate phase, 𝐾𝑠 is the heat conduction coefficient of the particles, and 𝛾𝑠 is the particle collision
energy dissipation term:
𝑘𝑠 =
3 𝜕
[ (𝛼 𝜌 𝑇 ) + ∇ ⋅ (𝛼𝑠 𝜌𝑠 V𝑠 𝑇𝑠 )]
2 𝜕𝑡 𝑠 𝑠 𝑠
(2)
+ 2𝜌𝑠 𝛼𝑠2 𝑑𝑠 (1 + 𝑒) 𝑔0 √
= (−𝑃𝑠 [𝐼] + 𝜏𝑠 ) : ∇V𝑠 − 𝛾𝑠 + ∇ ⋅ (𝑘𝑠 ∇𝑇𝑠 ) − 3𝐶𝐷𝛼𝑠 𝑇𝑠 ,
where the left-hand side of the equation represents the
accumulated kinetic energy of the particles and the righthand side is obtained from the granular temperature convection transport equation. The first term on the right-hand
side is the granular stress deformation work, the second
term is the energy dissipation of the particles due to their
inelastic collision with the wall, the third term is the thermal
conductivity of the particles, and the last term represents the
effect of the energy dissipation of the fluid and solid phases.
2
150𝜌𝑠 𝑑𝑠 √𝜋𝑇𝑠
6
[1 + 𝛼𝑠 𝑔0 (1 + 𝑒)]
384 (1 + 𝑒) 𝑔0
5
𝛾𝑠 =
12 (1 − 𝑒2 )
𝑑𝑠 √𝜋
𝑇𝑠
𝜋
(3)
𝜌𝑠 𝛼𝑠2 𝑔0 √𝑇𝑠3 .
It can be seen from the granular temperature equation
that the solid particles maintain laminar flow while the turbulent liquid phase gains energy and then dissipates it through
inelastic collision with the wall surface. The turbulent particle
phase is separated from the single-particle fluctuation zone
by the single particle fluctuation characteristics produced
by the inelastic collision of the randomly moving particles
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(c)
Figure 7: Dynamic pressure nephrogram under different entrance of turbulent kinetic energy of U-type tube channel.
with the wall. The turbulent fluctuation of the particle phase
reflects the turbulent fluctuation behavior of the solid particles. The particle temperature is determined by the single
particle level, and the turbulent kinetic energy of the particles
and the granular temperature may therefore be reduced by
the dissipation of turbulent kinetic energy. The ultimate dissipative state particle temperature is irreversibly transformed
into heat, which determines the particle temperature in a
thermodynamic sense.
3. Development of Physical Model of
U-Type Tube
3.1. Energy Conservation Equation. In this study, a U-type
tube was used as the object of the numerical thermodynamic
analysis of abrasive flow polishing. The tube is shown in
Figure 2(a). Based on the characteristics of a two-phase
solid-liquid abrasive flow polishing channel determined by
simulation, the actual channel geometry model of a U-type
tube requires simplification. The simplified geometry model
is shown in Figure 2(b).
3.2. Energy Conservation Equation. Because the abrasive
flow polishing medium in a U-type tube contains a viscous
fluid, investigation of the heat transfer during the process
requires an analysis of the flow field parameters, including
the dynamic pressure and the turbulent kinetic energy of
the machined parts near the wall. The structure of the
inner channel of a U-type tube is relatively simple; using an
unstructured hexahedral mesh, the mesh division is as shown
in Figure 3.
The flow of abrasive flow machining is complex and
involves strong fluctuations between laminar and turbulent
flows. The numerical simulation of the turbulent flow is more
vulnerable to the grid than that of the laminar flow. To ensure
accurate turbulence calculations, the area of the boundary
layer where the average flow changes quickly and the average
stress is large is selected as the near-wall region. To deal
with the calculation mesh, a boundary layer is added to the
part of the flow region near the wall, and the flow region
is divided into specific regions between the surface mesh
and the internal mesh. The grid independence check has
been done. After grid independence check, U-type model
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(c)
(d)
Figure 8: Turbulent kinetic energy cloud of U-type tube channel for different temperatures.
is divided into 541896 nodes, with 530796 hexahedral cells,
1581589 quadrilateral interior faces, and 17622 quadrilateral
wall faces. The grid volume is positive.
Based on the characteristics of abrasive flow machining,
ICEM, which is the preprocessing module of FLUENT, was
used in this study to develop the geometry of the flow region,
set the boundary type, and generate the meshes on the parts
of the U-type tube. A noncoupled implicit double precision
solver was used for the numerical analysis of the turbulent
RNG k-𝜀 model of the two-phase solid-liquid turbulent flow.
An abrasive flow media carrier was used as the main phase
and silicon carbide particles with a volume score of 0.3 mm
were used as the secondary phase. The boundary conditions
and the inlet and outlet velocities were set, and the rest of the
wall was considered to be solid. The effects of gravity were
taken into consideration.
After setting the parameters, the SIMPLEC algorithm
was used to solve the dynamic two-phase flow equation and
perform the calculations in the flow area. After initialization,
iterative computations were used to analyze the abrasive flow
machining process in the U-type tube. Using the calculation
results, we obtained the residual monitoring curve and some
numerical simulation figures.
4. Results and Discussion
The simulated U-type tube had an internal diameter of 4 mm
and the first introduction of turbulent kinetic energy was
under a condition different from that of a U-type tube
used for abrasive flow machining. For an inlet velocity of
70 m/s and initial temperature of 300 K, we analyzed the
turbulent kinetic energy in the near-wall region, as well as
the turbulence intensity, turbulent viscosity, and dynamic
pressure nephrogram. Figure 4 shows the images of the
turbulent kinetic energy in the U-type tube for different
inlet conditions. Figure 5 shows the turbulence intensity at
the corners and near the tube exit, and it can be observed
that the values are relatively large near the wall and reach a
maximum at the corner. The U-type tube channel abrasive
flow has a polishing effect at the bend in the tube where
the irregular motion of the abrasive grains is more intense.
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1.08e + 03
1.01e + 03
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1.03e + 03
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6.92e + 02
6.26e + 02
5.59e + 02
4.92e + 02
4.26e + 02
3.59e + 02
2.92e + 02
(a)
1.73e + 03
1.67e + 03
1.61e + 03
1.56e + 03
1.50e + 03
1.44e + 03
1.38e + 03
1.27e + 03
1.21e + 03
1.15e + 03
1.10e + 03
1.04e + 03
9.80e + 02
9.23e + 02
8.65e + 02
8.08e + 02
7.50e + 02
6.93e + 02
6.35e + 02
5.78e + 02
5.20e + 02
4.63e + 02
4.05e + 02
3.48e + 02
2.90e + 02
(b)
1.58e + 03
1.53e + 03
1.47e + 03
1.42e + 03
1.37e + 03
1.31e + 03
1.26e + 03
1.20e + 03
1.15e + 03
1.10e + 03
1.04e + 03
9.89e + 02
9.35e + 02
8.81e + 02
8.28e + 02
7.74e + 02
7.20e + 02
6.66e + 02
6.12e + 02
5.58e + 02
5.04e + 02
4.50e + 02
3.96e + 02
3.43e + 02
2.89e + 02
(c)
(d)
Figure 9: Turbulence intensity images of U-type tube channel for different temperatures.
Figures 4–7 show the turbulent kinetic energy. (a), (b), and
(c), respectively, correspond to 3.375, 9.375, and 13.5 m2 /s2 .
It can be observed from Figure 6 that the turbulent
viscosity near the wall of the channel was less than that at the
centre and that near the wall in the bend was larger than that
at the inner wall. The region of variable turbulent viscosity is
relatively large because the abrasive flow into the channel has
a microgrinding effect on the wall surface, and this increases
the mean temperature near the wall. The decrease in the
abrasive flow viscosity near the wall is greater than that at the
center owing to the effects of eddy diffusion.
As can be seen from Figure 7, the dynamic pressure of the
U-tube channel near the wall was very small, whereas those
at the inlet and outlet at the corners were relatively large. For
different inlet turbulent kinetic energy conditions near the
wall, the distributions of the turbulence intensity, turbulent
viscosity, and dynamic pressure can be divided into three
sections, namely, the straight section after the inlet (section
1), the curved section (section 2), and the straight section
before the outlet (section 3). In-depth comparative analyses of
the distributions for different inlet turbulent kinetic energies
near wall, turbulence intensities, turbulent viscosities, and
dynamic pressures were conducted using a velocity of 70 m/s
and initial temperature of 300 K. The distribution of the
turbulent kinetic energy and turbulence intensities in the Utype tube are given in Table 1.
An analysis of the data in Table 1 reveals that the turbulent
kinetic energies and turbulence intensities in the different
sections of the U-type tube increase with increasing inlet
turbulent kinetic energy. It can also be observed from the
table that, moving from section 1, through section 2, and
to section 3, the turbulent kinetic energy and turbulence
intensity increase and then decrease again. Comparative
analyses reveal that, for an inlet turbulent kinetic energy
of 9.375 m2 /s2 , there was better uniformity in the turbulent
kinetic energy and turbulence intensity in the tube channel.
Table 2 gives the distributions of the turbulent viscosity and
dynamic pressure in the tube for a velocity of 70 m/s and
initial temperature of 300 K.
It can be observed from Table 2 that the turbulent viscosity initially increased with increasing inlet turbulent
kinetic energy and then decreased. Moving from section 1,
Advances in Mechanical Engineering
9
4.06e + 00
3.89e + 00
3.73e + 00
3.57e + 00
3.41e + 00
3.25e + 00
3.08e + 00
2.92e + 00
2.76e + 00
2.60e + 00
2.44e + 00
2.28e + 00
2.11e + 00
1.95e + 00
1.79e + 00
1.63e + 00
1.47e + 00
1.30e + 00
1.14e + 00
9.80e − 01
8.18e − 01
6.56e − 01
4.94e − 01
3.32e − 01
1.71e − 01
8.64e − 03
4.01e + 00
3.84e + 00
3.68e + 00
3.51e + 00
3.34e + 00
3.18e + 00
3.01e + 00
2.84e + 00
2.68e + 00
2.51e + 00
2.34e + 00
2.18e + 00
2.01e + 00
1.84e + 00
1.68e + 00
1.51e + 00
1.34e + 00
1.18e + 00
1.01e + 00
8.42e − 01
6.75e − 01
5.08e − 01
3.42e − 01
1.75e − 01
8.46e − 03
(a)
(b)
3.98e + 00
3.82e + 00
3.66e + 00
3.50e + 00
3.34e + 00
3.19e + 00
3.03e + 00
2.87e + 00
2.71e + 00
2.55e + 00
2.39e + 00
2.23e + 00
2.07e + 00
1.91e + 00
1.76e + 00
1.60e + 00
1.44e + 00
1.28e + 00
1.12e + 00
9.61e − 01
8.03e − 01
6.44e − 01
4.85e − 01
3.26e − 01
1.67e − 01
8.32e − 03
3.97e + 00
3.81e + 00
3.64e + 00
3.48e + 00
3.31e + 00
3.14e + 00
2.98e + 00
2.81e + 00
2.65e + 00
2.48e + 00
2.32e + 00
2.15e + 00
1.99e + 00
1.82e + 00
1.66e + 00
1.49e + 00
1.33e + 00
1.16e + 00
9.99e − 01
8.34e − 01
6.69e − 01
5.03e − 01
3.38e − 01
1.73e − 01
8.19e − 03
(c)
(d)
Figure 10: Turbulent viscosity cloud of U-type tube channel for different temperatures.
Table 1: Distributions of turbulent kinetic energy and turbulence intensity in U-type tube for velocity of 70 m/s and initial temperature of
300 K.
Inlet turbulent kinetic
energy (m2 /s2 )
3.375
9.375
13.5
Turbulent kinetic energy (m2 /s2 )
section 1
section 2
section 3
150
160
171
285
290
308
through section 2, and to section 3, the turbulent viscosity
initially decreased and then increased. An examination of the
dynamic pressure data in the table reveals that the dynamic
pressure in the tube gradually decreased with increasing
inlet turbulent kinetic energy, and that the dynamic pressure
initially increased and then decreased with movement along
the tube. Comparative analyses revealed that there was
greater uniformity in the turbulent viscosity and dynamic
pressure distributions for an inlet turbulent kinetic energy of
9.375 m2 /s2 .
Using an inlet turbulent kinetic energy of 9.375 m2 /s2 and
initial velocity of 70 m/s, U-type channel simulations were
235
245
257
section 1
Turbulence intensity
section 2
section 3
1 000
1 140
1 200
1 240
1 370
1 442
1 102
1 240
1 302
performed to evaluate the process for different distributions
of the near-wall turbulence kinetic energy, turbulence intensity, turbulent viscosity, and dynamic pressure. In Figures 8–
11, (a), (b), (c), and (d), respectively, correspond to temperatures of 290, 300, 310, and 320 K.
It can be seen from Figures 8 and 9 that the outlet
turbulent kinetic energy and turbulence intensity near the
wall around the bend of the tube were relatively high and
attained maximum values at the bend. This indicates that
the abrasive flow polishing mainly occurred around the bend
of the U-type tube channel, which is conducive for light
processing.
10
Advances in Mechanical Engineering
7.88e + 06
7.56e + 06
7.25e + 06
6.93e + 06
6.62e + 06
6.30e + 06
5.99e + 06
5.67e + 06
5.36e + 06
5.04e + 06
4.73e + 06
4.41e + 06
4.10e + 06
3.78e + 06
3.47e + 06
3.15e + 06
2.84e + 06
2.52e + 06
2.21e + 06
1.89e + 06
1.58e + 06
1.26e + 06
9.46e + 05
6.31e + 05
3.15e + 05
1.37e + 02
7.77e + 06
7.45e + 06
7.13e + 06
6.80e + 06
6.40e + 06
6.15e + 06
5.83e + 06
5.51e + 06
5.18e + 06
4.86e + 06
4.53e + 06
4.21e + 06
3.89e + 06
3.56e + 06
3.24e + 06
2.92e + 06
2.59e + 06
2.27e + 06
1.94e + 06
1.62e + 06
1.30e + 06
9.72e + 05
6.48e + 05
3.24e + 05
2.28e + 02
(a)
(b)
7.69e + 06
7.30e + 06
7.00e + 06
6.77e + 06
6.46e + 06
6.15e + 06
5.85e + 06
5.54e + 06
5.23e + 06
4.92e + 06
4.62e + 06
4.31e + 06
4.00e + 06
3.69e + 06
3.38e + 06
3.08e + 06
2.77e + 06
2.46e + 06
2.15e + 06
1.85e + 06
1.54e + 06
1.23e + 06
9.23e + 05
6.16e + 05
3.08e + 05
3.49e + 02
7.83e + 06
7.52e + 06
7.20e + 06
6.89e + 06
6.58e + 06
6.26e + 06
5.95e + 06
5.64e + 06
5.32e + 06
5.01e + 06
4.70e + 06
4.39e + 06
4.07e + 06
3.76e + 06
3.45e + 06
3.13e + 06
2.82e + 06
2.51e + 06
2.19e + 06
1.88e + 06
1.57e + 06
1.25e + 06
9.40e + 05
6.27e + 05
3.13e + 05
1.42e + 02
(c)
(d)
Figure 11: Dynamic pressure nephrogram of U-type tube channel for different temperatures.
Table 2: Distributions of turbulent viscosity and dynamic pressure in U-type tube for velocity of 70 m/s and initial temperature of 300 K.
Inlet turbulent kinetic
energy (m2 /s2 )
3.375
9.375
13.5
Turbulent viscosity (kg/m⋅s)
section 1
section 2
section 3
1.2
1.32
1.18
2.15
2.22
2.16
It can be observed from the images shown in Figure 10
that the change in the turbulent viscosity at the outer wall near
the bend of the tube was greater than that at the outlet.
Figure 11 also reveals that the dynamic pressure of the
U-type tube near the wall was very small, whereas those
at the inlet and outlet were relatively large. The change in
temperature had little effect on the dynamic pressure distribution. Based on the distributions of the near-wall turbulent
kinetic energy, turbulence intensity, turbulent viscosity, and
dynamic pressure for different temperatures, the U-type tube
channel can be divided into three sections, namely, the
straight section after the inlet (section 1), the curved section
(section 2), and the straight section before the outlet (section
3). The distributions of the near-wall turbulent kinetic energy
2.8
2.82
2.81
Dynamic pressure (Mpa)
section 1
section 2
section 3
5.3
5.02
4.9
5.7
5.35
5.28
6.86
6.48
6.43
and turbulence intensity obtained by further comparative
analyses for different temperatures using an inlet velocity of
70 m/s and turbulent kinetic energy of 9.375 m2 /s2 are given
in Table 3.
As can be seen from Table 3, the turbulent kinetic energy
and turbulence intensity decreased drastically with increasing
temperature. This was because the decrease in the viscosity
of the abrasive flow medium with increasing temperature
caused the velocity gradient of the liquid phase to decrease,
thereby decreasing the turbulence intensity and turbulent
kinetic energy. The turbulent kinetic energy and turbulence
intensity initially increased between sections 1 and 2 and
then decreased between sections 2 and 3. The distributions
are, however, not uniform for any temperature. Table 4 gives
11
250
200
150
100
50
290
80
70
295
300
305
Tem
310
pera
315 320
ture
(K)
60
l
Ve
50
t
o ci
m
y(
/s)
Turbulent kinetic energy (m2 /s2 )
Turbulent kinetic energy (m2 /s2 )
Advances in Mechanical Engineering
400
350
300
250
200
290
80
70
295
300
305
Tem
310
pera
315 320
ture
(K)
50
l
Ve
ty
o ci
(m
/s)
(b)
Turbulent kinetic energy (m2 /s2 )
(a)
60
350
300
250
200
150
290
80
70
295
300
305
Tem
310
pera
315 320
ture
(K)
60
50
l
Ve
ty
o ci
(m
/s)
(c)
Figure 12: Relationship between the temperature and turbulent kinetic energy in a U-type tube ((a), (b), and (c) resp., correspond to sections
1, 2, and 3 of the tube).
Table 3: Distribution of turbulent kinetic energy and turbulence intensity in U-type tube channel for inlet velocity of 70 m/s and turbulent
kinetic energy of 9.375 m2 /s2 .
Temperature (K)
290
300
310
320
Turbulent kinetic energy (m2 /s2 )
section 1
section 2
section 3
192
330
280
160
290
245
145
280
230
134
272
220
section 1
1 190
1 140
1 110
1 090
Turbulence intensity
section 2
1 430
1 370
1 345
1 330
section 3
1 290
1 240
1 210
1 190
Table 4: Distributions of turbulent viscosity and dynamic pressure in U-type tube channel for inlet velocity of 70 m/s and turbulent kinetic
energy of 9.375 m2 /s2 .
Temperature (K)
290
300
310
320
section 1
1.47
1.32
1.26
1.16
Turbulent viscosity (kg/m⋅s)
section 2
section 3
2.47
2.42
2.22
2.82
2.21
2.81
2.16
2.46
section 1
5.04
5.02
5.01
5.01
Dynamic pressure (Mpa)
section 2
5.36
5.35
5.35
5.34
section 3
6.5
6.48
6.48
6.48
12
Advances in Mechanical Engineering
2.83e + 02
2.72e + 02
2.60e + 02
2.45e + 02
2.37e + 02
2.25e + 02
2.14e + 02
2.02e + 02
1.91e + 02
1.79e + 02
1.65e + 02
1.56e + 02
1.45e + 02
1.33e + 02
1.22e + 02
2.25e + 02
1.10e + 02
9.55e + 01
8.73e + 01
7.57e + 01
6.42e + 01
5.27e + 01
4.12e + 01
2.97e + 01
1.62e + 01
6.65e + 00
3.67e + 02
3.52e + 02
3.37e + 02
3.22e + 02
3.07e + 02
2.92e + 02
2.77e + 02
2.62e + 02
2.47e + 02
2.32e + 02
2.17e + 02
2.02e + 02
1.87e + 02
1.72e + 02
1.57e + 02
1.42e + 02
1.27e + 02
1.12e + 02
9.70e + 01
8.20e + 01
6.70e + 01
5.20e + 01
3.70e + 01
2.20e + 01
7.05e + 00
(a)
4.56e + 02
4.38e + 02
4.19e + 02
4.00e + 02
3.81e + 02
3.63e + 02
3.44e + 02
3.25e + 02
3.07e + 02
2.88e + 02
2.69e + 02
2.51e + 02
2.32e + 02
2.13e + 02
1.94e + 02
1.76e + 02
1.57e + 02
1.38e + 02
1.20e + 02
1.01e + 02
8.22e + 01
6.35e + 01
4.48e + 01
2.61e + 01
7.36e + 00
(b)
5.50e + 02
5.25e + 02
5.05e + 02
4.60e + 02
4.37e + 02
4.15e + 02
3.92e + 02
3.69e + 02
3.47e + 02
3.24e + 02
3.02e + 02
2.79e + 02
2.56e + 02
2.34e + 02
2.11e + 02
1.88e + 02
1.66e + 02
1.43e + 02
1.21e + 02
9.80e + 01
7.54e + 01
5.25e + 01
3.02e + 01
7.60e + 00
(c)
(d)
Figure 13: Turbulent kinetic energy cloud in U-type tube channel for different velocities.
the distributions of the turbulent viscosity and dynamic
pressure in the U-type tube channel for an inlet velocity of
70 m/s and turbulent kinetic energy of 9.375 m2 /s2 .
It can be observed from Table 4 that the turbulent viscosity in the U-type tube channel increased with decreasing
abrasion temperature and that the turbulent viscosity initially
decreased and then increased with movement from section
1, through section 2, and to section 3. It can also be seen
from Table 4 that there are smaller differences between the
turbulent viscosities in the different sections for an initial
temperature of 300 K. Examination of the dynamic pressure
data in Table 4 reveals a decrease with increasing temperature, although the decrease is not substantial.
The inlet turbulent kinetic energies for different conditions of the abrasive flow polishing in a U-type tube obtained
by simulation using different temperature conditions are
as given in Tables 1, 2, 3, and 4 for an inlet velocity of
70 m/s. It can be observed that the turbulent kinetic energy
varies with the temperature in each section of the tube.
Furthermore, using different inlet velocities of 50, 60, and
80 m/s, the relationship between the temperature and the
turbulent kinetic energy for each section of the tube was
obtained as shown in Figure 12. Figures 12(a), 12(b), and 12(c),
respectively, correspond to sections 1, 2, and 3.
It can be seen from Figure 12 that the turbulent kinetic
energy increased with increasing processing temperature in
each section of the tube. This was because the increasing
velocity increased the turbulent kinetic energy. The turbulent
flow was maximum at the corner of section 2, which indicates
that the abrasive flow movement was most intense in this
section, and the processing capacity was thus highest.
For comparison, numerical simulations of the polishing
in the tube were carried out using different inlet velocities
but the same inlet turbulent kinetic energy of 9.375 m2 /s2
and temperature of 300 K. The turbulent kinetic energy, turbulence intensity, turbulent viscosity, and dynamic pressure
were determined. The results are shown in Figures 13 to
16, where (a), (b), (c), and (d), respectively, correspond to
velocities of 50, 60, 70, and 80 m/s.
Figures 13 and 14, respectively, show the turbulent kinetic
energy and turbulence intensity in the U-type tube for
different velocities. As can be seen, the turbulent kinetic
energies and turbulence intensities near the wall of the tube,
at the inlet, near the bend, and at the outlet all increased with
Advances in Mechanical Engineering
13
1.70e + 03
1.64e + 03
1.55e + 03
1.51e + 03
1.45e + 03
1.39e + 03
1.33e + 03
1.27e + 03
1.21e + 03
1.14e + 03
1.08e + 03
1.02e + 03
9.58e + 02
8.97e + 02
8.35e + 02
7.73e + 02
7.11e + 02
6.49e + 02
5.88e + 02
5.26e + 02
4.64e + 02
4.02e + 02
3.40e + 02
2.79e + 02
2.17e + 02
1.50e + 03
1.45e + 03
1.39e + 03
1.34e + 03
1.29e + 03
1.23e + 03
1.18e + 03
1.13e + 03
1.07e + 03
1.02e + 03
9.64e + 02
9.10e + 02
8.56e + 02
8.02e + 02
7.49e + 02
6.95e + 02
6.41e + 02
5.57e + 02
5.33e + 02
4.80e + 02
4.26e + 02
3.72e + 02
3.18e + 02
2.64e + 02
2.11e + 02
(a)
1.85e + 03
1.82e + 03
1.75e + 03
1.68e + 03
1.61e + 03
1.54e + 03
1.47e + 03
1.40e + 03
1.33e + 03
1.26e + 03
1.19e + 03
1.12e + 03
1.05e + 03
9.84e + 02
9.15e + 02
8.45e + 02
7.76e + 02
7.07e + 02
6.37e + 02
5.68e + 02
4.99e + 02
4.29e + 02
3.60e + 02
2.91e + 02
2.22e + 02
(b)
2.06e + 03
1.98e + 03
1.91e + 03
1.83e + 03
1.75e + 03
1.68e + 03
1.60e + 03
1.52e + 03
1.45e + 03
1.37e + 03
1.29e + 03
1.22e + 03
1.14e + 03
1.07e + 03
9.89e + 02
9.13e + 02
8.36e + 02
7.60e + 02
6.84e + 02
6.07e + 02
5.31e + 02
4.54e + 02
3.78e + 02
3.02e + 02
2.25e + 02
(c)
(d)
Figure 14: Turbulence intensity image of U-type tube channel for different velocities.
increasing inlet velocity, and the values were maximum at
the bend of the tube, which indicates that the abrasion of the
grains was more intense at the bend.
It can be seen from the turbulent viscosity images in
Figure 15 that the turbulent viscosity near the wall of the Utube in section 1 was lower than that at the center.
Figure 16 shows that the dynamic pressure near the wall
of the tube increased with increasing velocity, although the
increase was not substantial. Based on the observed turbulent
kinetic energies, turbulence intensities, turbulent viscosities,
and dynamic pressures for the different velocities, the Utype tube channel can be divided into three sections, namely,
the straight section after the inlet (section 1), the curved
section (section 2), and the straight section before the outlet
(section 3). By in-depth comparative analyses of the different
parameters for the different inlet velocities using an inlet
temperature of 300 K and inlet turbulent kinetic energy
of 9.375 m2 /s2 , the turbulent kinetic energy and turbulence
intensity distributions presented in Table 5 were obtained.
It can be seen from Table 5 that the turbulent kinetic
energy and turbulence intensity in the U-type tube channel
increased with increasing velocity and that the turbulent
kinetic energy at the bend in the tube was relatively large.
The increase in the amplitude of the turbulent kinetic energy
was maximum for a velocity of 70 m/s. Moreover, the increase
in the turbulent kinetic energy and the increase in the
turbulence intensity between sections 1 and 2 and between
sections 2 and 3 for a velocity of 60 m/s were less than those
for velocities above 70 m/s. Hence, the distributions of the
two parameters for an inlet velocity of 70 m/s are preferable
for a U-tube channel. Table 6 gives the distributions of the
turbulent viscosity and dynamic pressure in the U-type tube
for an inlet temperature of 300 K and turbulent kinetic energy
of 9.375 m2 /s2 .
It can be seen from Table 6 that, for a high inlet velocity,
the turbulent viscosity in the U-type tube initially decreases
between sections 1 and 2 and then increases between sections
2 and 3. Comparative analyses revealed smaller differences
in the turbulent viscosity between sections 1 and 2 and
between sections 2 and 3 for an inlet velocity of 70 m/s.
Comparison of the dynamic pressures in Table 6 reveals that,
with increasing velocity, the dynamic pressure in the U-type
tube initially decreased between sections 1 and 2 and then
increased between sections 2 and 3. The turbulence viscosity
and dynamic pressure distributions for a velocity of 70 m/s
are thus preferable.
14
Advances in Mechanical Engineering
2.89e + 00
2.77e + 00
2.65e + 00
2.53e + 00
2.41e + 00
2.29e + 00
2.17e + 00
2.05e + 00
1.93e + 00
1.81e + 00
1.69e + 00
1.57e + 00
1.45e + 00
1.33e + 00
1.21e + 00
1.09e + 00
9.68e − 01
8.48e − 01
7.26e − 01
6.07e − 01
4.87e − 01
3.67e − 01
2.47e − 01
1.26e − 01
6.00e − 03
3.38e + 00
3.24e + 00
3.10e + 00
2.96e + 00
2.82e + 00
2.67e + 00
2.53e + 00
2.39e + 00
1.25e + 00
2.11e + 00
1.97e + 00
1.83e + 00
1.69e + 00
1.55e + 00
1.41e + 00
1.27e + 00
1.13e + 00
9.90e − 01
7.09e − 01
5.69e − 01
4.26e − 01
2.88e − 01
1.48e − 01
7.20e − 03
(a)
3.81e + 00
3.65e + 00
3.49e + 00
3.33e + 00
3.18e + 00
3.02e + 00
2.86e + 00
2.70e + 00
2.54e + 00
2.38e + 00
2.22e + 00
2.07e + 00
1.91e + 00
1.75e + 00
1.59e + 00
1.43e + 00
1.27e + 00
1.12e + 00
9.57e − 01
7.99e − 01
6.41e − 01
4.82e − 01
3.24e − 01
1.65e − 01
6.88e − 03
(b)
4.19e + 00
4.02e + 00
3.85e + 00
3.67e + 00
3.50e + 00
3.32e + 00
3.15e + 00
2.97e + 00
2.80e + 00
2.62e + 00
2.45e + 00
2.27e + 00
2.10e + 00
1.93e + 00
1.75e + 00
1.58e + 00
1.40e + 00
1.23e + 00
1.05e + 00
8.97e − 01
7.04e − 01
5.30e − 01
3.55e − 01
1.81e − 01
6.36e − 03
(c)
(d)
Figure 15: Turbulent viscosity cloud in U-type tube channel for different velocities.
Table 5: Distributions of turbulent kinetic energy and turbulence intensity in U-type tube channel for inlet temperature of 300 K and turbulent
kinetic energy of 9.375 m2 /s2 .
Velocity (m/s)
50
60
70
80
Turbulent kinetic energy (m2 /s2 )
section 1
section 2
section 3
110
235
195
125
255
210
160
290
245
180
315
260
section 1
1 080
1 100
1 140
1 160
Turbulence intensity
section 2
1 300
1 330
1 370
1 395
section 3
1 180
1 201
1 240
1 255
Table 6: Distributions of turbulent viscosity and dynamic pressure in U-type tube channel for inlet temperature of 300 K and turbulent
kinetic energy of 9.375 m2 /s2 .
Velocity (m/s)
50
60
70
80
section 1
0.8
1.05
1.32
1.35
Turbulent viscosity (kg/m⋅s)
section 2
section 3
1.8
2.5
2
2.65
2.22
2.82
2.4
3.08
section 1
5
5.01
5.02
5.02
Dynamic pressure (MPa)
section 2
5.34
5.35
5.35
5.36
section 3
6.49
6.49
6.48
6.51
Advances in Mechanical Engineering
15
5.78e + 06
5.54e + 06
5.30e + 06
5.06e + 06
4.82e + 06
4.58e + 06
4.34e + 06
4.10e + 06
3.86e + 06
3.62e + 06
3.37e + 06
3.13e + 06
2.89e + 06
2.65e + 06
2.41e + 06
2.17e + 06
1.93e + 06
1.69e + 06
1.45e + 06
1.21e + 06
9.64e + 05
7.23e + 05
4.82e + 05
2.41e + 05
1.04e + 02
4.03e + 06
3.86e + 06
3.60e + 06
3.53e + 06
3.36e + 06
3.19e + 06
3.02e + 06
2.85e + 06
2.69e + 06
2.52e + 06
2.35e + 06
2.18e + 06
2.02e + 06
1.85e + 06
1.68e + 06
1.51e + 06
1.34e + 06
1.18e + 06
1.01e + 06
8.40e + 05
6.72e + 05
5.04e + 05
3.36e + 05
1.68e + 05
5.32e + 01
(a)
7.85e + 06
7.52e + 06
7.20e + 06
6.87e + 06
6.54e + 06
6.22e + 06
5.89e + 06
5.56e + 06
5.23e + 06
4.91e + 06
4.58e + 06
4.25e + 06
3.93e + 06
3.60e + 06
3.27e + 06
2.94e + 06
2.62e + 06
2.29e + 06
1.96e + 06
1.64e + 06
1.31e + 06
9.82e + 05
6.54e + 05
3.27e + 05
1.56e + 02
(b)
1.02e + 07
9.81e + 06
9.38e + 06
8.96e + 06
8.53e + 06
8.10e + 06
7.68e + 06
7.25e + 06
6.82e + 06
6.40e + 06
5.97e + 06
5.54e + 06
5.12e + 06
4.69e + 06
4.27e + 06
3.84e + 06
3.41e + 06
2.99e + 06
2.56e + 06
2.13e + 06
1.71e + 06
1.28e + 06
8.53e + 05
4.27e + 05
2.32e + 02
(c)
(d)
Figure 16: Dynamic pressure nephrogram of U-type tube channel for different velocities.
By simulation of a U-type tube abrasive flow polishing
process using different inlet turbulent kinetic energies and
velocities, the data in Tables 1, 2, 5, and 6 were obtained
and were used to determine the relationship between the
velocity and the turbulent kinetic energy in each section of
the tube for an initial temperature of 300 K. Corresponding
relationships were obtained for initial temperatures of 290,
310, and 320 K. Figures 17(a), 17(b), and 17(c), respectively,
show the relationships for sections 1, 2, and 3.
Figure 17 shows that during the process of the abrasive
flow polishing in the U-type tube, the process velocity
increased, the turbulent kinetic energy in each section of
the tube increased, and the temperature of each section
decreased. As can be seen from Figure 18, the turbulent flow
was maximum in section 2. This was because the abrasive flow
changed suddenly by collision of the grains with the tube wall,
which resulted in increased turbulent kinetic energy.
The results of the simulation of the U-type tube abrasive flow polishing using different temperature and velocity
conditions were compared and analyzed based on the data
in Tables 3, 4, 5, and 6. An inlet turbulent kinetic energy
of 9.375 m2 /s2 , a process velocity of 70 m/s, and an initial
temperature of 300 K were used to determine the best distributions of the near-wall turbulent kinetic energy, turbulence
intensity, turbulent viscosity, and dynamic pressure. Initial
temperatures of 290, 310, and 320 K were similarly used for
the simulation, and the relationship between the abrasive flow
polishing temperature and velocity was determined. Inlet
turbulent kinetic energies of 3.375 and 13.5 m2 /s2 were further
used for the simulation and the corresponding relationships
were obtained. The obtained relationships are shown in
Figure 18. As can be seen from the figure, the process velocity
gradually increased with increasing temperature, but the rate
of increase began to reduce after attainment of a particular
temperature. At this point, the abrasive flow machining
velocity also almost ceased to increase. This was because
the abrasive flow fluid viscosity in the tube decreased with
decreasing temperature. There was no obvious effect of the
velocity change near the wall on the process.
5. Conclusion
In this study, we investigated abrasive flow polishing in a
U-type tube by an analysis of the thermodynamic energy
250
200
150
100
50
290
80
70
295
300
305
Tem
310
pera
315 320
ture
(K)
60
l
Ve
50
t
o ci
m
y(
/s)
Turbulent kinetic energy (m2 /s2 )
Advances in Mechanical Engineering
Turbulent kinetic energy (m2 /s2 )
16
400
350
300
250
200
290
80
70
295
300
305
Tem
310
pera
315 320
ture
(K)
50
l
Ve
ty
o ci
(m
/s)
(b)
Turbulent kinetic energy (m2 /s2 )
(a)
60
350
300
250
200
150
290
80
70
295
300
305
Tem
310
pera
315 320
ture
(K)
60
l
Ve
50
ty
o ci
(m
/s)
(c)
Figure 17: Relationship between velocity and turbulent kinetic energy in U-type tube ((a), (b), and (c) respectively show the relationships for
sections 1, 2, and 3).
Velocity (m/s)
100
80
60
40
320
20
0
310
5
10
300
15
290
mp
Te
K)
e(
tur
era
Figure 18: Relationship between temperature and velocity during abrasive flow polishing in U-type tube.
balance equation of the two-phase solid-liquid flow. The flow
and heat transfer characteristics were examined by numerical
simulation, and the effects of the turbulent kinetic energy,
turbulence intensity, turbulent viscosity, and dynamic pressure on the quality of the polishing process were examined to
establish the theory of the process.
Using different inlet conditions for the simulation, the
turbulent kinetic energies for different temperatures were
analyzed, and the relationships between the temperature and
the turbulent kinetic energy in the different sections of the
U-type tube during the abrasive flow polishing process were
obtained. By comparison of the turbulent kinetic energies
for different inlet velocities, the relationships between the
velocity and the turbulent kinetic energy in the different
sections of the tube were also determined. Furthermore,
by comparing the temperatures for different velocities, the
relationships between the temperature and the velocity for
different inlet turbulent kinetic energies were determined.
Advances in Mechanical Engineering
The parameter relationships obtained in this study can be
used for the optimization of the parameters of abrasive
flow polishing. Based on the obtained relationship between
the temperature and the velocity, the particle velocity and
temperature of abrasive flow polishing in a U-type tube can
be limited within an optimal range to improve the efficiency
of the process and the quality of the polished surface.
17
[11]
[12]
Conflict of Interests
The authors declare that there is no conflict of interests
regarding the publication of this paper.
[13]
Acknowledgments
[14]
The authors would like to thank the National Natural
Science Foundation of China (no. NSFC 51206011), Jilin
Province Science and Technology Development Program of
Jilin Province (no. 20130522186JH), and Doctoral Fund of
Ministry of Education of China (no. 20122216130001) for
financially supporting this research.
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