MEASUREMENT SCIENCE REVIEW, 16, (2016), No.

2, 87-95

MEASUREMENT SCIENCE REVIEW

ISSN 1335 - 8871 Journal homepage: http://www.degruyter.com/view/j/msr

The Effects of Fluid Viscosity on the Orifice Rotameter

Wei-Jiang1, Tao-Zhang1, Ying-Xu1,Huaxiang-Wang1 ,Xiaoli-Guo1, Jing-Lei1, Peiyong-Sang2
1
School of Electrical Engineering & Automation, Tianjin University, Tianjin, 300072, China
Tianjin Key Laboratory of Process Measurement and Control, Tianjin, 300072, China
2
Flow Measurement Center of Aviation Industries of China,Xinxiang,453049,China
Corresponding author: Ying-Xu E-mail: xuying @tju.edu.cn

Due to the viscous shear stress, there is an obvious error between the real flow rate and the rotameter indication for measuring viscous
fluid medium. At 50 cSt the maximum error of DN40 orifice rotameter is up to 35 %. The fluid viscosity effects on the orifice rotameter
are investigated using experimental and theoretical models. Wall jet and concentric annulus laminar theories were adapted to study the
influence of viscosity. And a new formula is obtained for calculating the flow rate of viscous fluid. The experimental data were analyzed
and compared with the calculated results. At high viscosity the maximum theoretical results error is 6.3 %, indicating that the proposed
measurement model has very good applicability.

Keywords: Rotameter, viscosity, experiment, regional model, flow measure.

1. INTRODUCTION designed and the designed rotameter flow equation was

In a typical variable area flow meter [1], the orifice derived by Urata [5]. Through experiments he found that the
rotameter has the advantages of simple structure, high outflow coefficient of the newly designed rotameter fell
reliability, convenient maintenance, low pressure loss, etc. within a wide range of Reynolds numbers (minimum to 50)
and is widely used in industrial flow measurement. When that remained constant, showing that viscosity has less
measuring a viscous fluid such as fossil oil, lube, beverages, influence on this rotameter. Vallascas [6] developed the
and dairy products, an obvious indication error OCCURS due magnetic suspension rotameter. An electric solenoid was
to the viscous shear stress effect. At 50 cSt the maximum installed in the outer conical tube to keep the float in a fixed
DN40 orifice rotameter error can be up to 35 %. It is position. The magnetic force was generated by the
therefore important and practical to study the fluid viscosity interaction of the variable magnetic field between the float
effects on the orifice rotameter. magnetism and the electric solenoid current. The current
Early research focused mainly on the rotameter output is proportional to the volume flow. Liu et al. [7]
measurement principle. Schoenborn and Colburn [2] first developed a capacitance sensor for the rotameter. The
began to deduce the rotameter flow equation. They thought simulation and practical flow experiments were carried out
that the rotameter could be regarded as a variable cross- with air as the medium. Baker [8], [9] researched the
section orifice flow meter. According to this analogy the influence of each production process on the rotameter
relationship between the flow rate and the differential measurement performance. Sondh et al. [10], [11] designed
pressure was directly obtained. The rotameter differential 3 different float shapes: a cone frustum, cone frustum with
pressure was derived based on the Bernoulli equation. A hemispherical base, and cone frustum with hemispherical
rotameter flow equation with the same Schoenborn form base and parabolic apex. A compensation algorithm based
was then obtained by Whitwell, Plumb, and Polentz [3], [4] on the BP neural network by Ning et al. [12] was used to
through solving the continuity equation with differential eliminate the temperature drift in metal tube rotameter using
pressure. an off-line training method based on the virtual instrument.
Modern research is concentrated mainly on two aspects, The computational fluid dynamics (CFD) method can be
first is the low rotameter accuracy in industrial applications. used to calculate the pressure, velocity distribution and other
In order to improve the rotameter measurement precision aspects of the data in the flow field, and also design the
and expand the measurement range, the rotameter structure product structural parameters. The German scholars Bueckle
was optimized by redesigning the float shape and strictly and Durst [13], [14] first introduced the CFD method into
controlling the production process. A cone float was rotameter research and exploited the Laser Doppler
_________________
DOI: 10.1515/msr-2016-0012

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Anemometer (LDA) to verify the results. It was concluded accurately analyze the viscous friction and pressure
that the simulation results were consistent with the LDA test difference, determine the flow field distribution and analyze
results. The reasons for the differences between the the fluid viscosity effects on the rotameter.
numerical and experimental data were analyzed. Their
research showed that the CFD method could analyze the 2. MODELING WITH WALL JET AND CONCENTRIC ANNULUS
rotameter internal flow field accurately and efficiently. LAMINAR THEORY
As seen from the above, in the structural optimization Viscous fluid flows from the bottom to the top of the float.
aspect, researchers mainly adopted the CFD method to Acting on the float force is the pressure drag Fp, buoyancy
optimize the float structure. The optimized results were Fρ, gravity G, and the viscous friction Fτ, respectively.
verified experimentally. The force balance equation:
The other aspect is the fluid viscosity effect on
measurement precision. A lot of research work has been
done [15], [16], with several viscosity correction schemes FP  F  G - F (1)
put forward in this aspect. Fisher [17] first proposed a
design that ignored the viscous effects, but the weight of the In the above formula, G   flV fl g , F  V fl g ,  fl is the
designed float was too small, and could not be applied to
practical production. A series of special float shapes that did density of the float, V fl is the volume of the float, ρ is the
not need to correct under certain viscosities were designed density of the fluid, and calculated directly. Differential
by Miller [18]. pressure and viscous friction can be calculated indirectly. As
From the data available for small rotameters that use shown in Fig.1. to calculate the differential pressure and
spherical floats in gas flow, Levin and Escorza [19] found a viscous friction accurately, the flow field is divided into 3
linear relationship for variable volumetric flow Q, density ρ, regions. Region A is the area from the bottom of the float to
and viscosity μ at a constant float height. At low Reynolds the bottom of the orifice. The flow field of region A is
numbers (Re < l), Qμ became a constant; while at high influenced mainly by the tube wall and the float, less
Reynolds numbers (Re > 2000), Qμ1/2 became a constant. affected by the orifice plate. Region B is the zone between
This method can be used to calibrate a gas rotameter the orifice plate and float, which has the maximum velocity
indirectly after the density and viscosity of the fluid has and maximum pressure gradient in the whole flow field.
been determined. Region C is the flow area through the orifice, which is also
Assuming that flow coefficient in the flow equation was a the confined annular wall jet region.
simple function of the Reynolds number and the viscosity
coefficient was replaced by the Reynolds number to
characterize the viscosity change, Wojtkowiak [20] obtained
a nonlinear equation for the flow rate with the float heights
through rotameter flow experimental data at different float
heights. The viscosity flow to theoretical flow ratio and the
viscosity correction curve were then obtained.
h

The study of viscosity effects on rotameter measurement β

can be divided into two categories. The first category is k
based on the existing float type flowmeter. The viscosity
r2 r1
correction curve is determined through experimental and
theoretical analysis, and thus the flow conversion
relationship between different viscosities is obtained. The
second category is based on the float structural design. The
design is verified and improved through experimentation to
eliminate the viscosity effect.
The simulation or experimental method is generally
adopted to optimize the float design and correct the float Fig.1. The orifice rotameter.
viscosity. Analysis and research into the flow field
distribution and interaction between the fluid and float is 2.1. Calculation of region A
lacking. The correction curve can be suitable only for a As shown in Fig.1. r2 is the float bottom radius, r 1 is the
specific caliber. Rotameters with different calibers need to tube radius. The float structure has axial symmetry; the flow
fit many different curves, so this method lacks universality.
is axisymmetric, so   0 . The main flow direction is z. The
The optimal design depends on the designer's experience 
and the cost of experimental verification is high with a long
flow is steady flow, so u  0 .
cycle. The concentric annulus laminar flow and wall jet t
theory were therefore adopted to study the flow field The Laminar concentric annulus solution is [21]
distribution and the float viscous shear stress. The flow field
was divided into three regions according to the relative
1 dp 2 r 2  r ( z )2 r
position of the float and the orifice. Viscous friction and u (r1  r 2  1 ln( )) (2)
pressure drop were calculated respectively in each region to 4 dz ln(r1 / r ( z )) r1

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 (r 2  r ( z )2 )2 dp  Q2 1 1 (12)
pAB  
r1
Q 2 rudr  [(r14  r ( z )4 )  1 ] (3) (
S B2 S A2
)
r( z) 8 ln(r1 / r ( z )) dz 2

r(z) = r2 + ztanβ . β is the float cone angle.

The pressure drops in region A are calculated first.

dp 8 Q (4)

dz (r 2  r ( z ) 2 ) 2
 [(r14  r ( z ) 4 )  1 ]
ln(r1 / r ( z ))

r1 r1 r1
1 1 1 (5)
r1 r( z) 1 r(z) 1 r(z) 2(r1  r ( z ))
ln  2{  ( )3  ( )5  }
r( z) r 3 1 r 5 1 r (r1  r ( z )) Fig.2. Region B.
1
1 1 1
r( z) r( z) r( z)

Pressure drop and friction in region B are calculated as

Substitute (5) into (4) follows. Region B could be divided into 2 areas as shown in
Fig.3. The pipe walls of region B2 do not vary with the
16 Q change of z. The surfaces of the pipe in region B1 changes
dp  dz (6)
 (r1  r ( z ))(r1  r ( z ))3 with the z coordinate. The change rule is

Set L as the length of float in region A, so there is r1'  r1  k  ( z  L) tan  (13)

L 16Q
pA   dz Compared with region A, there is just tube wall surface
0  (r1  r ( z ))(r1  r ( z ))3 change at z direction in region B1, but the friction and
4Q 1 r1  r2  L tan  r1  r2 L tan  (2r1  2r2  L tan  )
(7)
 [ ln( ) pressure drop form do not change. Equation (13) is
 r1 tan  2r12 r1  r2 r1  (r2  L tan  ) (r1  (r2  L tan  ))2 (r1  r2 )2 substituted into (7).
L tan 
 ]
(r1  (r2  L tan  ))(r1  r2 )r1 H h 16Q
pB1   dz
L [r1'  r ( z )][r1'  r ( z )]3 (14)
The viscous friction on the float is then calculated in 16Q H  h 1
 L ( J1  zJ 3 )( J 2  zJ 4 )3
 dz
region A, differentiate (2)

u r dp 1 dp 1 r12  r ( z )2 Here h is the float displacement in the current flow. H is

  (8)
r 2 dz r dz 4 ln(r1 / r ( z ) the length from float bottom to the top of the flow bevel, as
shown in Fig.1. set
The frictional resistance is
J1  r1  r2  k  L tan  (15)
u dp (r  r ( z ))2 r ( z ) (9)
   [ 1  ]
r dz 8r ( z ) 2 J 2  r1  r2  k  L tan  (16)

Substitute (4) into 9),

J 3  tan   tan  (17)
L 16Qr ( z ) (r  r ( z )) r (z) 2
F  2  [ 1  (10)
0  (r1  r ( z ))(r1  r ( z ))3 8r ( z ) 2
]dz
J 4   tan   tan  (18)

Simplification of the result is Through (14) the pressure drops in region B could be
calculated as
2 Q 4L tan  1 r 2  (r2  L tan  )2 (11)
F A  [  ln( 1 )] 16 Q
tan  (r1  r2 )(r1  (r2  L tan  )) r1 r12  r22 H h 1
 L ( J1  zJ 3 )( J 2  zJ 4 )3
pB1  dz
(19)
This is the float fluid friction in region A. The direction is 16  Q
 [ I1 ( H  h)  I1 ( L)]
along the float wall. 

2.2. Calculation of region B Function I1 (z) is defined as

As shown in Fig.2. a front step exists between region A J 2  zJ 4
and B, the radial height is k. The pressure drop between J32 ln
J1  zJ 3 J3 1 (20)
I1 ( z)   
region A and B can be directly obtained using the Bernoulli ( J1 J 4  J 2 J 3 )3 ( J1 J 4  J 2 J 3 )2 ( J 2  zJ 4 ) 2( J1 J 4  J 2 J 3 )( J 2  zJ 4 )2
equation,

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Also the friction of B1 area could be obtained by The calculation method is similar to region B. Pressure
substituting (13), (15) ~ (18) into (10) drop in region C is

F B1  Q[ I 2 ( H  h)  I 2 ( L)] (21) 16  Q H S 1 (24)

pC 
  H h (c1  zc3 )(c2  zc4 )3
dz

Function I2 (z) is defined as Here

J  zJ 4 (22) c1  2r2  b0  ( H  h)[tan(   )  tan  ]
2 4
I2 ( z)  ln 2 
J1 J 4  J 2 J 3 J1  zJ 3 J 4 ( J 2  zJ 4 ) (25)

The pressure drop in B2 is calculated by the Bernoulli c2  b0  ( H  h)[tan(    )  tan  ] (26)

equation, as follows
c3  tan(   )  tan  (27)
 Q2 1 1 (23)
pB2  (  )
2 SB22 SB21
c4  tan(    )  tan  (28)
As shown in Fig.2. SB1is the annulus area at the junction of
B1 and B2. SB2 is the annulus area at the outlet of the wall b0  rf  r2  ( H  h) tan  , is the width of the jet outlet.
jet. The friction in B2 is calculated with the friction in region There is
C. The reason will be presented in the following section.
16 Q H S 1
 H  h (c1  zc2 )(c3  zc4 )3
pC  dz
2.3. Calculation of region C (29)
As fluid flows through the gap between the orifice plate 16 Q
 [ I1 ( H  S )  I1 ( H  h)]
and the float into the area of C, the annular wall jet is 
formed as shown by the dotted lines in Fig.3. with the
diffusion angle α of wall jet [22]. I1 (z) function is defined by (20).
The friction force of region C cannot be calculated using
the divergent tube model. This is because the flow field in
region C is limited by the wall jet, the maximum velocity of
the profile um is closer to the wall compared with the
diverging tube and the viscous friction force on the float is
larger.
According to the wall jet theory the maximum velocity of
the profile um and the outlet velocity u0 have the following
relationship [23],

um b
 3.50 0 (30)
Fig.3. The wall jet in region C. u0 z

In Fig.3. the lateral dotted line is formed by the points As shown in Fig.4. the ratio of the velocity at separation
whose velocities are zeros in the z direction, and could be point n of float boundary layer to the outlet m is calculated
regarded as a wall in the calculation of pressure drop. The according to (30), and the results are shown in the following
flow model in region C could be simplified as the concentric table.
annular diffusion flow as shown in Fig.4.
Table 1. Ratios of the velocity at n to outlet velocity at m.

h b0 (h+S)/b0 um/u0
0.0049 0.000738 22.9 0.73
0.0123 0.001471 16.52 0.86
0.0192 0.002154 14.49 0.92
0.0284 0.003065 13.18 0.96
0.0362 0.003837 12.56 0.99

The table shows that even at the farthest point away from
the jet outlet, the reduction of maximum velocity is very
Fig.4. Simplified model. small. The flow field in region C is confined by the wall jet,

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so the diffusion angle is less than the wall jet diffusion angle
[24], and the maximum velocity profile is larger than that of
the wall jet. It can be approximately considered that the
viscous friction on the float in region C is everywhere equal,
and equal to outlet friction. The total friction of region C
and B2 can be concluded.

4 Q(h  S ){rf  3[r2  ( H  h) tan  ]} (31)

F ,C  B2 
[rf  r2  ( H  h) tan  ][rf  r2  ( H  h) tan  ]2

2.4. Diffusion angle of the confined wall jet

As shown in Fig.6. the fluid jets from the nozzle outlet.
The entrainment of ambient fluid is produced in the upper Fig.6. The relationship of K and the diffusion angle α.
boundary as a free jet and this mixing force develops
downwards. The boundary layer is formed in the lower Equation 33 illustrates that the diffusion angle is inversely
boundary by the solid wall friction effect and develops proportional to the float limitation to the flow field.
upward with a potential core area in the center. According to
the maximum velocity of profile um, the flow field is divided 2.5. Theoretical flow rate
into two areas: the upper mixing zone and the lower According to the above derivation process, the friction and
boundary layer [25]. pressure drop in each region are obtained, so,
Verhoff [26] and Rajaratnam [27] presented the empirical
formula of the relationship between the maximum profile F  F A  F B1  F , B2 C (34)
velocity and the outlet velocity. Rajaratnam also studied the
entrainment velocity in the free mixing zone and the effects
of the surface roughness on the thickness of the boundary P  PA  PAB  PB1  PB2  PC (35)
layer.
F A, F B , F , B C are decided by (11), (21), (31),
1 2

respectively, and are the linear function of the flow rate Q;

PA , PB , PC are dominated by (7), (19), (29),
1

respectively, and are the linear function of Q;

PAB , PB are ruled by (12), (23), respectively, and are the
2

quadratic function of Q.
Substitute (34), (35) into the formula (1)

F  Af P  G  F

Quadratic equation of flow rate Q is obtained.

Fig.5. The confined wall jet. a1Q 2  a2Q  a3 (36)

The diffusion angle does not change with the variance in

the outlet velocity [28]. The diffusion angle is therefore a1 , a2 , a3 are the coefficient of equation.
determined by the outlet width and the cross-sectional area
of the concentric annular pipe after the outlet. Therefore the a1Q 2  A f (PAB  PB2 )
dimensionless coefficient K is defined to characterize the
Q 2 1 1 Q 2 1 1
degree of limitation. [29]  Af [ (  )  ( 2  2 )]
2 S B2 2
SA 2 S B1 S B2
b0 (32)
K
A1 Af  1 1 1 1
a1  ( 2 2 2  2 ) (37)
2 S B S A S B1 S B2
The diffusion angle of the wall jet is determined using
simulation (CFD). The relationship between the diffusion
angle α and K is then obtained. Similarly

α = 11.543 - 43.998K (33) a2Q  FA  FB1  F , B2 C  Af (PA  PB1  PC ) (38)

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a3  G  F . (39) 3.2. Experiments with viscous fuild

4050 aviation lubricant oil was used as the experimental
Finally the theoretical flow rate Q is obtained. medium. The viscosity range for this oil is from 10 cSt to
50 cSt. The device is composed of seven parts: a liquid
circulation system, test tube, static weighing system, start-
a2  a22  4a1a3
Q (40) stop equipment, electric heater, refrigeration unit and the
2a1 control system. Weighing method was used to calibrate the
rotameter. The valve was opened and the oil in the tank
3. EXPERIMENTAL RESEARCHES pumped out using the oil pump (Group). After the oil was
3.1. The experimental water flow device filtered the temperature was changed (heating or cooling),
the pressure stabilized and the oil circulated into the oil tank
The experimental water flow device at Tianjin University
through the commutator’s bypass pipeline. When the flow
Flow Laboratory was used to perform the calibration, as
rate and oil temperature met the requirements, the standard
shown in Fig.7. Water flow was applied to regulate the
scale was reset and the initial readings noted m0 (kg). The
pressure and the device flow rate was continuously adjusted.
commutator was started, allowing the oil to flow into the
Both the weighting and master meter methods were used
weighing container. This simultaneously triggered the timer
with the experimental device for verification. The weighing
and started the count. When the desired weighting value was
method uncertainty is 0.05 %, and that for the master meter
achieved, the commutator was started again so that the oil
method is 0.2 %.
could flow through the bypass line back into the tank. The
timer was stopped when the oil was back inside the tank.
The standard scale m (kg) and time t (s) values were
recorded. The average flow of qm through the rotameter can
be calculated using the following formula,

m  m0
qm  
t (41)
here
w
 ——The correction coefficient for air
w  a

buoyancy
Fig.7. Standard water device.
ρa——The density of air, kg/m3
1. Inlet Valve, 2. Filter Tank, 3. Master Meter, 4. Electric Control
Valve, 5. Surge Tank, 6. Excluding-ordure Valve, 7. Support Plate, ρw——The density of oil, kg/m3
8. Metal Tube Rotameter, 9. Clamping, 10. Flow Regulating Valve,
11. Nozzle, 12. Commutator, 13. Counter Tank, 14. Drain Valve,
15. Electronic Weigher, 16. Control Cabinet, 17. Computer
18, 19. pressure measuring hole.

qv0min, 0.25qv0max, 0.4qv0max, 0.7qv0max and qv0max are selected,

and weighting method is used to calibrate DN50 metal tube
rotameter. Measuring range is1~10 m³/h on the standard
device.
Fig.8. Principle of the variable viscosity device.
Table 2. The water flow device verification data.
The experimental device performances index is as follows:
the Indicating Experimental Error temperature change range from -35℃ to 150℃, temperature
h(mm) control accuracy within 1℃, flow range from 0.1 m3/h to
value (m³/h) flow rate(m³/h) (%)
4.90 1.00 1.01 1.20 80 m3/h, and expanded uncertainty (K=2) is about 0.05 %.
To avoid errors caused by float shake and parallax errors
13.20 2.50 2.51 0.24 in the calibration process, 4 ~ 20 mA output analog signals
19.40 4.00 3.98 -0.60 were converted and collected in the corresponding sampling
time for calculating the average flow rate.
28.90 7.00 7.07 0.97 The experimental procedure was as follows:
36.50 10.00 9.96 -0.36 1) The flow meter was calibrated in the variable viscosity
flow device. Each flow point was calibrated twice at
Data in Table 2. showed that the uncertainty of the positive and reverse range. The calibration process was as
rotameter reached 1.5 grade standard. follows: first adjust the frequency transformer to make the

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measured voltage value corresponding to the indication flow In the tables, qv0 is the rotameter indication flow rate, qv
rate of qv0min, 0.25qv0max, 0.4qv0max, 0.7qv0max and qv0max, represents the actual flow rate of the 4050 oil obtained using
respectively. The medium and flow rate temperatures were the weighing method and δ 1 is the full scale error:
recorded to calculate the average positive and reverse range
values. qv 0 - qv (42)
1   100 %
2) The lubricating oil was taken from the variable qmax
viscosity experimental system. The sample temperature was
kept the same as the lubricating oil temperature in the As the lubricating oil density is close to the density of
device. The lubricating oil viscosity was measured using a water, the medium density effects on the measurement can
NDJ-5S digital viscometer and the density was measured be ignored.
using a densimeter.
The experimental data indicate that at the same viscosity,
3) The medium temperature was changed to change the
the full scale error increases with the increase in flow rate.
medium viscosity and density. Steps (1) ~ (2) were then
repeated. This is because with the increase in flow rate, the boundary
layer thickness on the float and guide rod becomes thinner,
3.3. Experiment data and analysis making the velocity gradient in the boundary layer become
larger. The viscous force on the float and guide rod becomes
The experimental data are shown in Table 3. - Table 6. larger from the increasing velocity gradient in the boundary
layer.
Table 3. The experimental data for v =9.99 cSt.
At the same indicated value, the full scale error becomes
larger with the increase in medium viscosity. This is because
qv0/m³/h t/℃ ρ/kg/m³ v/cSt qv/m³/h δ1/% the boundary layer thickness increases and the flow area is
1.03 67.10 970.12 10.23 0.97 0.52 reduced as the viscosity increases. At the same time the
2.65 66.78 970.11 10.33 2.46 1.95 viscous force is increased with the increase in medium
3.95 66.85 970.12 10.30 3.28 6.64 viscosity. The full scale error therefore becomes larger due
to the double resistance effect.
7.09 68.83 970.05 9.75 6.59 4.94
9.89 70.55 969.97 9.33 8.33 15.62 4. CALCULATION RESULTS AND ERROR ANALYSIS
Table 4. The experimental data for v =30.86 cSt. Equation (40) is the flow rate formula considering the
viscous friction. The calculated results according to (40) and
the experimental error are shown in Fig.9. - Fig.12.
qv0/m³h t/℃ ρ/kg/m³ v/cSt qv/m³/h δ1/%
1.07 33.88 971.50 31.18 0.82 2.51
2.52 33.98 971.50 31.08 2.06 4.59
4.13 34.13 971.50 30.93 3.01 11.22
7.05 34.35 971.48 30.65 5.25 18.01
9.87 34.55 971.48 30.45 7.33 25.40

Table 5. The experimental data for v =38.13 cSt.

qv0/m³/h t/℃ ρ/kg/m³ v/cSt qv/m³/h δ1/%

1.01 28.23 971.80 38.28 0.69 3.24
2.55 28.48 971.78 37.85 2.00 5.45
4.00 28.38 971.78 38.00 2.78 12.2 Fig.9. The calculated results for v = 10 cSt.
7.01 28.18 971.78 38.35 4.86 21.45
9.96 28.33 971.80 38.15 7.14 28.24

Table 6. The experimental data for v =50.96 cSt.

qv0/m³/h t/℃ ρ/kg/m³ v/cSt qv/ m³/h δ1/%

1.05 22.00 972.10 52.53 0.63 4.19
2.56 21.63 972.13 53.68 1.92 6.41
4.01 22.08 972.10 52.28 2.68 13.36
7.01 23.20 972.01 48.88 4.82 21.92
9.91 23.70 972.00 47.43 6.47 34.47 Fig.10. The calculated results for v = 30 cSt.

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 MEASUREMENT SCIENCE REVIEW, Volume 16, No. 2, 2016

It is a new way that introduces wall jet and concentric

annulus laminar theories to studying the influence of
viscosity. A viscous fluid flow mathematical model through
the orifice was established. And equation (40) is a new
formula to calculate the flow rate at high viscosity. In the
laminar flow the maximum error is only 6.3 %, indicating
that the model has good laminar flow applicability.
In the turbulence state, because separation is produced at
the corner of the orifice plate and the bottom and top of the
float, the differential pressure spatial distribution
consequently changes. Model calculation for differential
Fig.11. The calculated results for v = 40 cSt. pressure based on the laminar flow will generate a large
error. On this occasion, relative to the viscous friction, the
differential pressure plays a main role. The flow rate should
7 be directly calculated using the classical rotameter equation.
Experimental data

6 Calculated results

5
NOMENCLATURE
A1= The cross-sectional area of concentric annular pipe
Q(m^3/h)

3 after the outlet, m2

2 Af = The area at the maximum cross section of float, m2
1 b0 = The width of outlet, m
0 c1, c2, c3, c4= Constants associated with the structure of the
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 float
h(m)
Fp = The differential pressure force, N
Fig.12. The calculated results for v = 50 cSt. Fρ = Buoyancy, N
Fτ= Viscous friction, N
Fig.9. demonstrates that at low viscosity all calculated G = Gravity, N
results are larger than the experimental data. The error H = The length from float bottom to the top of bevel, m
increases with the increase in flow rate. The calculation h= The distance of float moving in the current flow, m
result shows that when v=10 cSt, the flow rate is 2.457 m3/h, I1(z), I2(z) = Function associated with z coordinate
the rotameter Re number is 1259.4 at the annulus gap and J1, J2, J3, J4 = Constants associated with the structure of the
the flow field in region C belongs to turbulence. When the float
flow field is in the turbulent state, the boundary layer K= Dimensionless coefficient characterization of the
separation phenomenon occurs, in which the boundary layer degree of limitation
separates from the float surface and the recirculation zone k = The radial length of front steps, m
will be produced at the largest float section and also at the L= The distance from bottom of the float to the bottom
corner of the orifice plate and pipe wall. The separation zone of the orifice, m
will seriously affect the boundary at the outflow region, m = The position of outlet
which will change the wake meiobar range. The viscosity n = The position at the separation point of boundary
effect cannot be considered to only be limited to a thin layer layer
of fluid near the surface. The pressure resistance calculation Δp = Differential pressure, Pa
is not only related to the flow field distribution in region C,
Q = Volume flow rate, m3/h
but also in connection with the separation state and the wake
qv0 = Indication flow rate of the rotameter, m3/h
region. Separation and turbulence will change the pressure
difference spatial distribution greatly. There will be larger qv = The actual flow rate, m3/h
errors between the experimental data and calculated results r1 = The tube radius, m
based on the laminar flow model. r1' = The tube radius of region B1, m
When v = 30 cSt, the flow rate is 7.235 m3 / h and Re is r2 = The float bottom radius, m
1111.4. Fig.10. - Fig.12. illustrate that the error between the rf = Orifice radius, m
theoretical calculations and experimental data is very small, S = The length of the cylinder at upper part of
indicating that the model is reasonable. rotameter, m
SA, SB, SB1, SB2=The annular area of the subscript, m2
5. CONCLUSION um = The maximum velocity of the section in mixing
The experimental and theory methods were used to zone, m/s
examine the viscosity effect on rotameter measurement in u0= Outlet velocity, m/s
this research. The experimental results indicate that at the α = Diffusion angle of the wall jet
same viscosity, the viscosity effect increases with the β = Float cone angle
increase in flow rate. The error also increases at the same θ= Rotation angle of column coordinates
flow rate. The greater the viscosity becomes, the larger the φ = Orifice cone angle
error produced. μ = Dynamic viscosity

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 MEASUREMENT SCIENCE REVIEW, Volume 16, No. 2, 2016

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