Tribology International 75 (2014) 31–38

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Tribology International
journal homepage: www.elsevier.com/locate/triboint

Numerical analysis of plain journal bearing under hydrodynamic

lubrication by water
Gengyuan Gao, Zhongwei Yin n, Dan Jiang, Xiuli Zhang
Shanghai Jiao Tong University, School of Mechanical Engineering, Shanghai 200240, China

art ic l e i nf o a b s t r a c t

Article history: The article aims to provide references for designing water-lubricated plain journal bearings. Considering
Received 28 August 2013 the differences between the physical properties of the water and of the oil, the effects of eccentricity
Received in revised form ratio on pressure distribution of water film are analyzed by computational fluid dynamics (CFD).
3 March 2014
Then numerical analysis of journal bearings with different dimensions is undertaken under different
Accepted 4 March 2014
Available online 12 March 2014
rotational speeds. Based on the analysis, a reference is produced for selecting the initial diameter
dimension which is used to design an efficient water-lubricated plain bearing under the given load and
Keywords: rotational speed. At last, the reference is verified by an experimental case.
Plain journal bearings & 2014 Elsevier Ltd. All rights reserved.
Water-lubrication
Cavitation
CFD

1. Introduction Christopherson [13] determined the pressure distribution by utilizing

the mathematical model of “relaxation”. Tayal et al. [14] investigated
Saving energy and reducing the amount of pollution released to the effect of nonlinearity on the performance of journal bearings
the environment have increasingly become the most concerned with finite width by using the finite element method.
issues in machine design [1]. Water-lubrication becomes a tendency There are two modeling approaches for hydrodynamic flow
for its convenience, green, safe and energy saving. The application of issue. On one hand, there are some models based on the Reynolds
water-lubricated journal bearings is widespread, such as shipbuild- equation [15–17]. On the other hand, there are models based on
ing, industrial machinery and equipment, transportation industry, the Navier–Stokes system of equations [18–20]. Recently, with the
food industry, and pharmaceutical industry [2–4]. So far, studies of rapid development of computer technology and hardware, many
water-lubrication are mainly about tribology behavior of sliding researchers used commercial computational fluid dynamics (CFD)
surfaces of cermet or polymer in water, only a few of them involve programs in their investigations [21–23]. Computer programs
bearing performances including load carrying capacity and friction by CFD provide a useful design service but are generally time
performances [5–8]. consuming and not every designer has access to them. Thus it is
The disadvantage of water as a lubricant is that its viscosity is necessary to draw on the results from such programs in order to
much lower than that of oil and grease [9]. It contributes to low develop the reference for designing plain bearings, which cover a
hydrodynamic load carrying capacity of a water-lubricated plain wide range of plain bearing geometries and operating conditions
bearing. Little research has been made to improve hydrodynamic [24]. The CFD package, FLUENT, is a suitable software for numerical
load carrying capacity of plain journal bearings. Most of the simulating and analyzing the flow problem.
researchers were greatly interested in investigating the theories Generally, the design procedure of plain journal bearings starts
and experiments of hydrodynamic lubrication and paid no attention with selecting the bearing dimensions, such as journal diameter
to how to improve hydrodynamic load carrying capacity of a journal (D), bearing length (L) and radial clearance (C). At first, the
bearing. Sivak and Sivak [10] obtained a numerical solution of the designer is more interested in load carrying capacity (W) of plain
Reynolds equation by the modified Ritz method and Sfyris and bearing which mainly depends on the basic dimension, namely
Chasalevris [11] achieved a path of obtaining the exact analytical journal diameter (D). However, there is no design reference for
solution of the Reynolds equation for the lubrication of journal selecting an initial diameter dimension of a water-lubricated plain
bearings with finite length. Kingsbury [12] determined the pressure journal bearing under hydrodynamic lubrication.
distribution by means of an experimental electrical analogy and One important design decision is the selection of length over
diameter ratio (L/D). It is obvious that a long bearing has a higher
load capacity in comparison to a shorter bearing. However,a
n
Corresponding author. long bearing increases the risk of bearing failure due to misalign-
E-mail address: yinzw@sjtu.edu.cn (Z. Yin). ment errors. In addition, a long bearing reduces the amount

http://dx.doi.org/10.1016/j.triboint.2014.03.009
0301-679X/& 2014 Elsevier Ltd. All rights reserved.
32 G. Gao et al. / Tribology International 75 (2014) 31–38

Nomenclature Re ,Rc mass transfer source terms connected to the growth

and collapse of the vapor bubbles, respectively
C radial clearance of bearing t time
D diameter of journal W load carrying capacity
e eccentricity of journal ρ fluid density
F cond condensation coefficient ρl liquid density
F evap evaporation coefficient ρv vapor density
hmin minimum film thickness ε eccentricity ratio
I unit tensor μ fluid viscosity
ε eccentricity ratio ϕ attitude angle of the journal
L bearing width θ angular coordinate
!
N rotational speed of journal v fluid velocity vector
P static pressure τ stress tensor
p local far-field pressure av vapor volume fraction
pb bubble surface pressure anue nucleation site volume fraction
RB bearing radius vv vapor phase velocity
RJ journal radius s liquid surface tension coefficient
Rb bubble radius

of fluid circulating in the bearing, which results in a higher peak The hydrodynamic pressure eventually terminates in the divergent
temperature inside the lubrication film and the bearing surface. A part of the gap, where the pressure might fall below the vapor
general rule-of-thumb is that the length over diameter ratio (L/D) pressure of water, and cavitation occurs. When steady state is
should be between 0.5 and 1.5 [25]. Short bearings (L/D between reached, the journal is displaced from the bearing with a center
0.5 and 0.7) are recommended for designing the conventional oil distance (e), which is referred to the journal eccentricity. The
and grease lubricated bearings. As mentioned, the limitation of eccentricity ratio (ε) and the radial clearance (C) are important
replacing oil and grease by water is its low viscosity. In order to measures of the load carrying capacity of the bearing. They also
improve load capacity of water-lubricated plain bearing, length provide the measure of the thickness of the lubricant film which
over diameter ratio (L/D) should be higher than that of the separates the journal and the bearing.
conventional oil and grease-lubricated bearings. In present work,
L/D is fixed to 1.0 for researching the performance of water- 3. Theoretical considerations
lubricated plain bearing.
Radial clearance (C) is an important design parameter for 3.1. Governing equation
determining the load carrying capability of plain bearing. Smaller
clearances generate higher load carrying capacity of a bearing for For all flows, FLUENT solves conservation equations for mass
the same operating conditions. However, misalignment, solid and momentum [27]. For flows involving heat transfer or com-
contaminants and roughness of the bearing surfaces pose some pressibility, an additional equation for energy conservation is
limitations on minimizing radial clearance (C). Experience over the solved. For flows involving species mixing or reactions, a species
years has resulted in a guide for most designers that radial conservation equation is solved, or, if the non-premixed combus-
clearance (C) for a bearing is taken as one thousandth of journal tion model is used, conservation equations for the mixture fraction
radius [26]. In this work, the performances of water-lubricated and its variance are solved. Additional transport equations are
plain bearings are studied under the radial clearance of one also solved when the flow is turbulent. In the present work, the
thousandth of the journal radius. conservation equations for laminar flow and cavitation model are
Except for journal diameter (D), bearing length (L) and radial presented based on the above assumptions.
clearance (C), eccentricity ratio (ε) plays an important role in load
carrying capacity of plain journal bearing. In order to study the 3.1.1. Mass conservation equation
reference for designing water-lubricated plain journal bearing, the The equation for conservation of mass, or the continuity equation,
paper investigates the relationship between eccentricity ratio (ε) can be written as
and distribution of pressure produced by hydrodynamic lubrication. ∂ρ !
In the present research three-dimensional CFD models are þ ∇ðρ v Þ ¼ 0 ð1Þ
∂t
developed, using the FLUENT package, to investigate hydrodynamic
performances of water-lubricated plain journal bearings. Then a
reference is produced for selecting the initial diameter dimension
which is used to design an efficient water-lubricated plain bearing
under hydrodynamic lubrication.

2. Plain journal bearing model

Fig. 1 shows the coordinate and the schematic of a simple plain

journal bearing in a steady-state configuration. The plain journal
bearing is submersed in water. The hydrodynamic action generates
dynamic pressure in water, primarily in the convergent part of the
journal-bearing gap, to counteract the load thereby separating the Fig. 1. Definition of the coordinate and the schematic of a simple plain journal
journal surface from the bearing surface with a thin lubricant film. bearing.
 G. Gao et al. / Tribology International 75 (2014) 31–38 33

Eq. (1) is the general form of the mass conservation equation and is 3.2. Assumptions and boundary conditions
valid for incompressible as well as compressible flows where ρ and
!
v are the fluid density and fluid velocity vector, respectively. 3.2.1. Assumptions
The following assumptions for the plain bearing model are
used in this article:
3.1.2. Momentum conservation equations A rigid aligned bearing with the geometry of Fig. 1 is considered; a
Conservation of momentum in an inertial (non-accelerating) steady state is assumed; the flow is laminar and water properties are
reference frame is described by constant; a constant external vertical load F is applied to the journal.
∂ ! !! ! !
ðρ v Þ þ ∇ðρ v v Þ ¼  ∇P þ ∇ðτÞ þ ρ g þ F ð2Þ
∂t 3.2.2. Boundary conditions
where P is the static pressure, τ is the stress tensor (described The governing equations are solved in steady state, taking no
! ! account of gravity force, and the operating pressure is set to
below), and ρ g and F are the gravitational body force and
external body forces (e.g., that arise from interaction with the 101,325 Pa. According to the dimensional parameters and the working
! condition of the journal bearings, the viscous model is set to the
dispersed phase), respectively. F also contains other model-
dependent source terms such as porous-media and user-defined laminar model and enhanced wall treatment. To simplify the geome-
sources. The stress tensor τ is given by try, one side of the water film clearance is used as an inlet and the
other as an outlet. The boundary conditions of the inlet and outlet are
τ ¼ μ½ð∇! ! !
T
v þ ∇ v Þ  2=3∇  v I ð3Þ respectively “pressure inlet” and “pressure outlet” with gauge pressure
at zero Pascal. The outer surface of the water film is modeled as a
where μ is the fluid viscosity, I is the unit tensor, and the second “stationary wall” and the inner surface is modeled as a “moving wall”
term on the right hand side is the effect of volume dilation. with an absolute rotational speed which equals the angular velocity of
the journal. The pressure-based solver is chosen for the numerical
analysis in the paper. A converged solution can be obtained more
3.1.3. Cavitation model quickly using SIMPLEC compared to SIMPLE [27]. Therefore, in the
When water film pressure falls below the saturation water vapor present work, the velocity-pressure coupling is treated using the
pressure, the liquid would rupture and cavitation occurs. SIMPLEC algorithm, and the others are default.
In cavitation, the liquid–vapor mass transfer (evaporation and
condensation) is governed by the vapor transport equation:
3.3. Analysis and results
∂
ðav ρv Þ þ ∇ðav ρv vv Þ ¼ Re Rc ð4Þ
∂t 3.3.1. Eccentricity ratio
where Re and Rc account for the mass transfer between the liquid Eccentricity ratio (ε) is an important factor on load capacity of
and vapor phases in cavitation. They are based on the Rayleigh– plain journal bearing. In order to study the relationship between
Plesset equation describing the growth of a single vapor bubble in eccentricity ratio (ε) and distribution of pressure produced by
a liquid as in the following equation: hydrodynamic lubrication, the paper analyzes the effect of the
different eccentricity ratio (ε) on pressure distribution of the


p  p 2s μ dRb
2
d Rb 3 dRb hydrodynamic water film, the parameters of the journal bearing
Rb þ ¼ b  4 l ð5Þ
dt 2 2 dt ρl ρl Rb ρl Rb dt used in the numerical analysis are shown in Table 1. In the present
study, water properties at 20 1C as listed in Table 2 are employed.
Neglecting the second-order terms and the surface tension force, Fig. 2 shows the pressure distribution vs. the eccentricity
Eq. (5) is simplified to ratio (ε). The red zones are the positive pressure zones formed
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi by hydrodynamic water film. Hydrodynamic performances of plain
dRb 2 pb p
¼ ð6Þ journal bearing mainly depend on the size and distribution of the
dt 3 ρl positive pressure on the zones. It is obvious that, with eccentricity
ratio (ε) increasing, the area of the positive pressure zones is
This equation provides a physical approach to introduce the
decreasing. A conclusion from Fig. 2 is that high eccentricity
effects of bubble dynamics into the cavitation model. It can also
be considered to be an equation for void propagation and, hence,
mixture density. In the present work, Zwart–Gerber–Belamri Model
Table 1
is applied. Assuming that all the bubbles in a system have the same
The parameters used in the numerical analysis.
size, the final form of this cavitation model is as follows:
If p r pv D 80 mm
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L 80 mm
3anue ð1  av Þρv 2 pv p C/RJ 1‰
Re ¼ F evap ð7Þ
Rb 3 ρl ε 0.4–0.9
N 1500 rpm
If p Z pv
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3av ρv 2 p  pv
Rc ¼ F cond ð8Þ
Rb 3 ρl
Table 2
Water properties at 20 1C.
where,
Saturation water vapor pressure 2340 Pa
Rb ¼ bubble radius ¼10  6 m. Saturation density of water 998.2 kg/m3
0.5542 kg/m3
anue ¼nucleation site volume fraction¼5  10  4 . Saturation density of water vapor
Dynamic viscosity of water 10  3 Pa s
F evap ¼evaporation coefficient ¼50. Dynamic viscosity of water vapor 1.34  10  5 Pa s
F cond ¼condensation coefficient ¼0.01.
34 G. Gao et al. / Tribology International 75 (2014) 31–38

Fig. 2. The film pressure distribution contour vs. the eccentricity ratio (ε) for (a) ε ¼ 0:4, (b) ε ¼ 0:5, (c) ε ¼ 0:6, (d) ε ¼ 0:7, (e) ε ¼ 0:8, and (f) ε ¼0.9.

ratio (ε) is not conducive to hydrodynamic performances of plain

journal bearing in running. Therefore, in the design phase, a
designer should keep in mind that an expected eccentricity ratio
(ε) of the journal in running should be a proper value for designing
an efficient water-lubricated journal bearing.
Fig. 3 shows the pressure distributions on the middle ring of the
journals for the different eccentricity ratios. It can also be observed
that the negative pressure is negligible compared to the positive
pressure. In addition, the positive pressure is significantly different
with different eccentricity ratio (ε). The range of the positive pressure
peak is narrow when the eccentricity ratio (ε) is 0.8 and 0.9.The
situation may cause stress concentration when the plain bearing is
running. When the eccentricity ratio (ε) is small, such as 0.4 or 0.5,
the pressure is too low to separate the journal surface from the
bearing surface. Therefore, 0.6 or 0.7 is regarded as a proper value for
eccentricity ratio in design stage of a plain bearing.
Fig. 3. The film pressure distribution in the circumferential direction.

3.3.2. Journal diameter used in the numerical analysis covers a wide range of 10–
In the present work, in order to develop a design reference for 100 mm while the rotational speed (N) of the journal varies from
selecting the initial diameter of a water-lubricated plain journal 100 rpm to 5000 rpm. According to experience over the years and
bearing under hydrodynamic lubrication, journal diameter (D) the previous analysis, the length over diameter ratio of 1.0, the
 G. Gao et al. / Tribology International 75 (2014) 31–38 35

Table 3
Numerical results of parameters charactering the analyzed bearings.

N D

10 20 30 40 50 60 70 80 90 100

100 2.2 9 20.1 35.7 55.9 80.5 109.6 143.2 181.2 210.4
200 4.4 17.9 40.3 71.6 111.9 161.2 219.5 286.9 363.3 422
300 6.7 26.8 60.4 107.5 168.2 242.2 330.1 431.5 546.7 635.2
400 9 35.8 80.7 143.6 224.6 323.8 441.1 576.9 731.2 849.2
500 11.2 44.7 100.6 178.9 279.8 403.4 549.9 719.2 911.8 1058
600 13.3 53.2 119.8 213.2 333.4 480.8 655.5 857.6 1087.5 1261.1
700 15.4 61.5 138.7 246.7 385.9 556.7 759.1 993.4 1260 1460
800 17.5 69.8 157.2 279.7 437.7 631.4 861.2 1127.4 1430.4 1656.7
900 19.5 77.9 175.4 312.2 488.7 705.2 962.1 1259.7 1598.7 1851.2
1000 21.5 85.9 193.5 344.5 539.3 778.3 1061.9 1390.8 1765.6 2044.2
1500 31.1 124.9 281.3 501.4 785.5 1134.6 1550 2032.4 2583.5 2995.2
2000 40.6 162.8 367 654.3 1026 1483.6 2028.7 2663.5 3390.9 3938
2500 49.9 200 451.3 805.3 1263.9 1828.9 2503.5 3291.1 4195.6 4879.4
3000 59.1 236.9 535 955.1 1500 2172.7 2977.5 3919.1 5003.3 5818.5
3500 68.2 273.7 618.2 1104.4 1735.6 2516.2 3452 4549.6 5813.7 6764.5
4000 77.4 310.2 701 1253.3 1971 2860.3 3928.5 5184.3 6630.6 7699.5
4500 86.4 346.6 783.9 1402 2206.6 3205.5 4408 5821.8 7452.1 8658
5000 95.4 383 866.5 1550.7 2442.9 3552 4890.8 6461.9 8278.5 9596.1

Fig. 5. Design reference for designers selecting initially diameter of a plain journal
Fig. 4. Design reference for designers selecting initially diameter of a plain journal
bearing hydrodynamic lubricated by water when eccentricity ratio ε¼ 0.7.
bearing hydrodynamic lubricated by water when eccentricity ratio ε¼ 0.6.

radial clearance of one thousandth of journal radius and eccen- are small, eccentricity ratio (ε) of 0.7 is preferred to design an
tricity ratio of 0.6 and 0.7 are chosen to investigate the perfor- efficient hydrodynamic plain bearing. Fig. 5 shows the reference
mance of water-lubricated plain bearings. when eccentricity ratio (ε) is 0.7.
In the work, numerical results of the film pressure distribution
are obtained by CFD package FLUENT, and one can find the
composition of fluid film forces, namely, the load carrying capacity 4. Experimental verification of a case
(W) of hydrodynamic water film, by integrating the pressure
distribution on the film area. The load carrying capacity (W, 4.1. Introduction of apparatus and test bearing
newton) of journal bearings with different diameters under
different journal rotational speeds are presented in Table 3 when Fig. 6 shows the schematic of the apparatus used for measuring
eccentricity ratio (ε) is 0.6. The diameter of the journal (D, mm) the thickness of water film and the friction coefficient (f) of journal
and rotational speed (N, rpm) are discrete. In order to draw a bearings under the given conditions.
common reference for selecting the initial diameter of a plain The main shaft is supported by two hydrostatic bearings
journal bearing hydrodynamic lubricated by water, a numerical and driven by a Siemens servo motor with a maximum speed of
treatment with the results is made by the function of counter fit in 6000 rpm and a maximum power of 22 kW. The desired load is
MATLAB. Then the design reference is shown in Fig. 4. When the applied using a hydraulic cylinder via a leading bar. As a case, the
work conditions of a plain bearing are in the range of the design test bearing is composed by the bushing and the sleeve. It is
reference, designers can select a proper diameter for the bearing. resistant to wear and corrosion. The shaft is made of hardened
For example, as shown in Fig. 4 (The dotted line), when the given 42CrMo steel with 0.30 Poisson's ratio, 206 GPa elastic modulus
rotational speed (N) is 3300 rpm, if 4500 N is required for the load which is resistant to wear and corrosion. It has an outer diameter
carrying capacity, the diameter of the plain journal bearing should 80.002 mm. The surface finish (Ra) is 0.2 μm consistently for the
be around 82 mm. shaft. The material of the sleeve is the same as the shaft and the
Of course, if the anti-fatigue and anti-wear properties of the bushing is made of PTFE, carbon fiber and PEEK. The bearing has
bearing material are distinguished and the misalignment errors an inner diameter 80.082 mm and an outer diameter 120.6 mm.
36 G. Gao et al. / Tribology International 75 (2014) 31–38

Fig. 6. Schematic of the apparatus (not to scale):1-servo motor; 2-damper; 3-coupling;

4-shaft;5-hydrostatic bearings; 6-test bearing; 7-sleeve; 8-hydraulic cylinder; 9-leading
bar; 10-measurement cell for friction coefficient; 11-measurement cell for thickness of
water film; 12-tank; and 13-base.

Fig. 7. Schematic of installation position of four displacement sensor in test bearing

(not to scale).

Fig. 8. The real object of measurement cell for thickness of water film.
The inner surface finish (Ra) is 0.4 μm consistently for the bushing.
The bearing length is 80.21 mm. Therefore, the length over
diameter ratio is 1.0 and the radial clearance is one thousandth by the eq. (9) and the results are shown in Table 5. As shown in
of the shaft radius. Table 4, the friction coefficient f are all less than 0.003. It can be
Fig. 7 shows the installation position of four displacement concluded that the test bearing is running under hydrodynamic
sensor in test bearing. The photo of real object is shown in lubrication according to the literature [28].
Fig. 8. As shown in Fig. 7, the clearance S0, between the inner face As shown in Table 5, the maximum is 0.0179 mm for thickness
of test bearing and the front of displacement sensor probe, is of water film and the minimum is 0.0140 mm. Since the inner
0.65 mm. The clearance d, between the outer face of journal and diameter of the test bearing is 80.082 mm, a conclusion is that the
the front of displacement sensor probe, can be measured by eccentricity ratio is about 0.6 (from 0.65 to 0.5525) when the test
four displacement sensor. So the thickness of water film h can be bearing is running under the given conditions.
drawn by the following equation: The initial diameter dimension of the test bearing is selected
from Fig. 4. Based on above discussion, when the test bearing is
h ¼ d  S0 ð9Þ
running under the given conditions, the test bearing is running
with about 0.6 in eccentricity ratio under hydrodynamic lubrication.
4.2. Experimental conditions Therefore, the reference in Fig. 4 should be proper for designing
efficient water-lubricated plain bearings under the given conditions.
The properties of water are shown in Table 2. The inner
diameter of the test bearing is 80.082 mm, in order to verify the
reference of Fig. 4 (eccentricity ratio ε ¼ 0.6), the rotational speeds
in the experiments are 1500 rpm, 2000 rpm, 2500 rpm, 3000 rpm, 5. Conclusion
3500 rpm, and 4000 rpm and the corresponding applied loads are
2050 N, 2700 N, 3350 N, 3950 N, 4850 N, and 5200 N respectively. A hydrodynamic analysis, based on three-dimensional CFD
In the experiments, if the test bearing is running with eccen- techniques, is developed and applied to hydrodynamic plain journal
tricity ratio ε ¼0.6 under hydrodynamic lubrication, the reference bearing lubricated by water. Taking account of the differences
in Fig. 4 should be proper for designing efficient water-lubricated between the physical properties of the water and of the oil, the
plain bearings. effects of cavitation on hydrodynamic lubrication are also considered
via a proper cavitation model. Eccentricity ratio (ε) is considered as
4.3. Experimental result and discussion an important factor for improving hydrodynamic performances of
plain journal bearings. A conclusion is that a higher or lower
Table 4 shows experimental results of the test bearings under eccentricity ratio, which is selected as an expected eccentricity ratio
the given conditions. The thickness of water film h can be drawn in running in the design phase, results in a bad hydrodynamic plain
 G. Gao et al. / Tribology International 75 (2014) 31–38 37

Table 4
Experimental results of the test bearings under the given conditions.

1500 rpm 2000 rpm 2500 rpm 3000 rpm 3500 rpm 4000 rpm
2050 N 2700 N 3350 N 3950 N 4850 N 5200 N

Friction coefficient f 0.00295 0.00278 0.00253 0.00237 0.00229 0.00221

d of No. 1 0.6674 0.6667 0.6663 0.6661 0.6649 0.6642
d of No. 2 0.6679 0.6675 0.6672 0.6671 0.6657 0.6647
d of No. 3 0.6664 0.6663 0.6659 0.6658 0.6647 0.6640
d of No. 4 0.6675 0.6672 0.6668 0.6667 0.6655 0.6646

Table 5
The results of thickness of water film.

1500 rpm 2000 rpm 2500 rpm 3000 rpm 3500 rpm 4000 rpm
2050 N 2700 N 3350 N 3950 N 4850 N 5200 N

h of No. 1 0.0174 0.0167 0.0163 0.0161 0.0149 0.0142

h of No. 2 0.0179 0.0175 0.0172 0.0171 0.0157 0.0147
h of No. 3 0.0164 0.0163 0.0159 0.0158 0.0147 0.0140
h of No. 4 0.0179 0.0172 0.0168 0.0167 0.0155 0.0146

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