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Flow Characteristics of V-Cone Flowmeter in Wet Steam Chao Wang, Tianyi Li, Hongbing Ding*, Jinxia Li Tianjin Key Laboratory of Process Measurement and Control School of Electrical and Information Engineering Tianjin University, Tianjin, China *
[email protected] Abstract—Wet steam is a common two-phase flow whose mass flow-rate can be measured by V-Cone flowmeter. The mass flow coefficient CD of V-Cone is no longer approximately constant which affects the measuring accuracy subject to the complex flow characteristics of wet steam and discrete droplets. Firstly, five main impact factors for CD of V-Cone were obtained by dimensional analysis, namely, vapor Reynolds number, liquid-vapor density ratio, mass loading ratio, potential to kinetic energy ratio and dimensionless droplet diameter respectively. Then CFD simulation was carried out to study the influence of these parameters. Eulerian multiphase model and SST k-Ȧ turbulence model were adopted for steady simulation. The simulation results coincided well with the experimental results. The results indicate that CD is almost not affected by vapor Reynolds number and potential to kinetic energy ratio; it increases with the liquid-vapor density ratio, mass loading ratio and dimensionless droplet diameter, among which mass loading ratio is the main factor. Keywords—V-Cone flowmeter; Wet steam flow; Dimensional analysis; Mass flow coefficient; CFD; Droplet characteristics;
I.
INTRODUCTION
As an important clean energy, steam is widely used in industrial process. Due to energy loss during the long-distant transport, superheated stream gradually condenses and then becomes wet steam. The accurate measurement and reasonable utilization have an important significance on the improvement of production efficiency and energy conservation [1]. However, as a vapor-liquid two-phase flow, wet steam has very complicated characteristic and is difficult for accurate metrology. In recent years, as a new differential pressure (DP) type flowmeter, V-Cone shows great potentiality in wet steam measurement for the merits of low pressure loss, lower requirement for straight pipe length and self-cleaning ability. The discharge coefficient and expansibility factor have always been the research topics of V-Cone for their significant effects on the metrology accuracy. For example, Dahlstrom et al. [2] found that the expansibility factor of V-Cone was somewhere in between orifice and venturi. Stewart et al. [3] drew the conclusion that Reynolds number and inner diameter of pipe had little effect on expansibility factor. Peters et al. [4] and Mclver et al. [5] concluded that V-Cone had high accuracy in a wide measuring range. Xu et al. [6]-[7] found that the discharge coefficient of V-Cone was related to the angle of cone and equivalent diameter ratio. He et al. [8]-[9] obtained the fitting formula of mass flow coefficient CD of V-Cone. However, the former studies on V-Cone mostly aimed at single-phase flow, and there has been lack of research on wet steam flow. In this study, five main impact factors were National Natural Science Foundation of China under Grant 61627803, 51506148 and 61673291, Natural Science Foundation of Tianjin under Grant 16JCQNJC03700 and 15JCYBJC19200.
978-1-5090-3596-0/17/$31.00 ©2017 IEEE
obtained for CD of V-Cone flowmeter in wet steam by dimensional analysis, namely, vapor Reynolds number, liquid-vapor density ratio, mass loading ratio, potential to kinetic energy ratio and dimensionless droplet diameter respectively. Then, CFD model was built to investigate the effects of these parameters on the values of CD, which had been validated by experiments. Finally, the detailed results about the effects of each parameter were obtained. II.
THEORY OF V-CONE FLOWMETER
A. Mass Flow Rate The structure and size of V-Cone flowmeter are shown in Fig. 1. The mass flow rate of V-Cone is defined by: qm =
Cε
π
1- β
4
4
D 2 β 2 2 Δp ρ
(1)
where qm is the mass flow rate, C is discharge coefficient, İ is expansibility factor, ȡ is the density of the fluid, D is the inner diameter of pipe, ǻp is the differential pressure, d is the largest cross section diameter of cone, ȕ is the equivalent diameter ratio described as: β = D 2 - d 2 D .
Fig. 1. Structure of V-Cone flowmeter.
For wet steam, the discharge coefficient C and expansibility factor İ of V-Cone flowmeter are always coupled together. Thus, the mass flow coefficient CD [9] is expressed by:
C D = Cε
(2)
B. Dimensional Analysis in Wet Steam TABLE I.
PARAMETERS AND THEIR SIGNS, UNITS AND DIMENSIONS. Parameter
Sign
Unit
Dimension
Differential pressure
ǻp
Pa
ML-1T-2
Mass flow rate of vapor and liquid
mg & mp
kg/s
MT-1
Kinetic viscosity of gas
ȝg
Pa·s
ML-1T-1
Density of vapor and liquid phase
ȡ g & ȡp
kg/m3
ML-3
Inner diameter
D
m
L
Front and back angle of cone
ș1 & ș2
rad
1
Gravity
g
m/s2
LT-2
System pressure
P
Pa
ML-1T-2
Diameter of cone
d
m
L
Droplet diameter
dp
m
L
According to π theorem, any of the variables Ȇi (i = 1~7) can be replaced by combining them with other variables Ȇj (j i). CD is obtained by:
CD = Cε = f (φ , Re sg ,
TABLE II.
DIMENSIONLESS GROUPS AND THEIR TRANSFORMATIONS.
Π Π1 =
Δp ρ m D 4 mm2
Π2 =
mp
ρp ρg
mg2
PD ρ g
Π7 =
=
=
dp D
mg
gD U sg2
P
ρ gU sg2
Π 5' =
(4)
Dimensional analysis just points out the possible impact factors rather than the detailed flow characteristics of V-Cone. Thus, both CFD and experiment were utilized to study effects of the dimensionless factors on CD of V-Cone flowmeter. III. CFD SIMULATION A. Geometric and Grid Model In view of measurement accuracy of V-Cone flowmeter, ȕ should not be too large under the condition of limited pressure loss [11]. The V-Cone size is shown in Fig. 1, where ȕ = 0.543, D = 50 mm, d = 42 mm, ș1 = 50°, ș2 = 101°.
=φ
1 = Re sg Π3
Π '4 = Π 4 =
4
mg2
Π 3' =
mg
g ρ g2 D 5
mp
ρp dp P , , ) 2 ρ g ρ gU sg D
For the rotational symmetry of cone body, we drew up half computational domain in two-dimensional simulation. The volume was meshed by structured grids, and grid refinements were conducted near the pipe wall and bluff body to properly capture the very thin boundary layer and improve solution accuracy. The local grid near the V-Cone is shown in Fig. 2.
1 8(1- β 4 ) = Cε = CD Π1 π 2 β 4 Π '2 = Π 2 =
μg D
Π4 =
Π6 =
Π1' =
=φ
mg
Π3 =
Π5 =
Π'
(3)
The effect of gravity was not obvious due to the droplets were too small, hence the parameter of Froude number Frg can be ignored, and Eq. (3) was simplified to CD = Cε = f (φ , Re sg ,
During the long-distant transport of wet steam, the dryness, pressure, temperature as well as density may change, and the influence of different physical properties on the mass flow coefficient CD can’t be characterized in a single parameter. For this reason, dimensional analysis was employed [10] to determine the possible impact factors on the CD of V-cone in wet stream. According to dimensional analysis, all of the parameters in the wet stream flow and their dimensions are listed in TABLE I, where mass (M), length (L) and time (T) are the basic dimensions.
ρp dp P , Frg , ˈ ) 2 ρg ρ gU sg D
ρp ρg
ρg
U 1 1 = sg Π5 Π4 -1 gD Π '6 = Π 6 =
Π '7 =
ρ p − ρg
Fig. 2. Local grid partition of two-dimensional model.
= Frg
P
ρ gU sg2
dp D
In wet stream, the liquid phase mainly exists in the form of droplet as a discrete phase, thus the liquid viscosity is ignored. In this paper, we don’t consider the influence of the cone structure; hence ș1, ș2 and d are constant for the given V-Cone. Accordingly, the number of the total parameter n is 10, the number of basic dimensions m is 3, and then the number of dimensionless groups equals 7, expressed as Ȇ1, Ȇ2 … Ȇ7. It should be noted that these seven dimensionless variables are mutually independent. The dimensionless groups of original forms and transformations are listed in TABLE II, where mm is the sum mass flow rate, and ȡm is the mixture density.
The mesh quality of boundary layer could be indicated by a dimensionless distance y+, which stands for the distance from the mass center of the first layer grid to the wall. y+ is defined by:
y+ =
yuτ
ν
,
u+ =
u , uτ
uτ =
τw ρ
(5)
where y is the distance to the wall, uτ is the friction velocity, v is the kinematic viscosity, u is the average velocity of fluid, τw is the wall shear stress. B. Governing Equations Eulerian model is adopted to simulate the wet stream flow and phase b should satisfy the following governing equations in the calculation. Continuity conservation equation:
n G ∂ (α b ρb ) + ∇ ⋅ (α b ρb ub ) = ¦ ( m ab − m ab ) + Sb ∂t a =1
(6)
where the subscript a is primary phase and b is secondary phases; Į is the volume fraction; u is the velocity of fluid; ȡ is the density; mab is the mass transfer from phase a to phase b; the source term Sb is zero. Momentum conservation equation: G GG G ∂ (α b ρbub ) + ∇ ⋅ (α b ρbubub ) = −α b∇P + ∇ ⋅ τ b + α b ρb g ∂t n G G G G G G + ¦ Rab + m abuab − m bauba + Fb + Flift , b + Fvm , b a =1
(
) (
)
(7)
where P is the pressure act on each phase; Fb is an external body force; Flift, b is a lift force; Fvm, b, is a virtual mass force; uab is the interphase velocity; Rab is an interaction force between phases a and b. Rab is defined by: n
G
¦R
ab
a =1
n G G αα ρ f = ¦ K ab ˄ua − ub˅ , K ab = b a a
τa
a =1
(8)
where Kab is the momentum exchange coefficient; f is the drag function; IJa is the particulate relaxation time. Drag force function and drag coefficient are defined by [12]: 24 (1 + 0.15 Re 0.687 ) Re r < 1000 C d Re r r ° (9) f = , Cd = ® Re r 24 ° 0.44 Re r ≥ 1000 ¯ G G ρ u −u d (10) Re r = a a b b
μb
where Rer is the particle Reynolds number; ȡa is the density of phase a; ȝb is the kinetic viscosity of phase a; db is the diameter of droplets of phase b. In addition, the turbulence model has a crucial influence on the precision and the accuracy of CFD simulation. SST k-Ȧ model is adopted for its combination of k-İ model used in area far away from the wall and k-Ȧ model used near the wall in order to get high resolution flow solution in line with physical significance. SST k-İ model is defined as follows:
μt = ∂ D( ρ k ) = ∂x j Dt
ρ a1 k max( a1ω , Ω F2 )
ª ∂k º ∂u * «( μ + σ k μ t ) » + τ ij i − β ρω k ∂x j »¼ ∂x j «¬
(11)
(12)
ª ∂ω º ∂u «( μ + σ ω μt ) » + γμtτ ij i − ∂x j »¼ ∂x j «¬ (13) ρ 1 ∂k ∂ω 2 βρω k + 2(1 − F1 ) σ ω 2 ω ∂x j ∂x j D ( ρω ) ∂ = ∂x j Dt
where ȝt is the eddy viscosity coefficient; k is the turbulent kinetic energy; Ȧ is the turbulence frequency; ȝ is the molecular viscosity coefficient; ui is the flow velocity component; xj is coordinate axis; ȍ is the vorticity; F is composite function; IJij is the viscous shearing stress; ȕ*, ȕ, Ȗ, ık, ıȦ, ıȦ2 and a1 are constant. C. Numerical Scheme The primary phase was saturated vapor and the secondary phase was water liquid. The heat transfer was not considered due to small change of temperature in short-distance flow. The density of vapor and water were calculated by IF–97 formula [13]. The inlet boundary condition was velocity inlet, and the outlet condition was pressure outlet. The cone body and tube wall were wall, and the midline of tube was axis for rotational symmetry. Second order upwind scheme for all spatial discretization was combined with the phase coupled SIMPLE scheme. Turbulence parameter was defined by turbulence intensity I and hydraulic diameter DH, where I = 0.16Re (-1/8). Diameter was set 4~14 ȝm [14], the calculation iterated in steady state and the residual criterion for convergence was 10-4. IV. EXPERIMENTAL VALIDATION A. Experimental Rig
Fig. 3. The experimental device.
The experimental device is shown in Fig. 3. Superheated steam was produced by steam boiler, and then flow through the self-operated pressure stabilizing valve into desuperheater and buffer tank. Wet steam was obtained after desuperheating and long-distant transport. The flow rate of wet steam was adjusted to achieve stable pressure and flow rate. The tested V-Cone flowmeter was installed in the downstream area where both thermal equilibrium and force balance were reached. The mass flow rate of wet steam through V-Cone flowmeter was measured by weighing the mass change of water in tank. The operating pressure can be up to 10 MPa; the range of mass flow rate is from 0 to 4 t/h, the highest working temperature is 300 ć. B. Comparison of Experiment and CFD The CFD and experimental results were compared in P = 4.6 MPa and P = 5.5 MPa cases respectively, and the relationship between mass flow coefficient CD and mass loading ratio φ is shown in Fig. 4. Under different pressure, CD will rise with the increase of φ. The maximum relative error between experimental and CFD results didn’t exceed 1.6%, which indicated a good quantitative agreement. Thus, it is basically reliable to study the flow characteristics of V-Cone and its impact factors in wet steam by this CFD
model.
0.89
0.90
P = 4.6 MPa
0.88 CD=0.87+5.2×10-4ρp/ρg
0.89
CD 0.87
0.88
0.86
CD 0.87
Experiment CFD
0.86
φ = 0.2 φ = 0.4
dry steam baseline
0.85 15
20
25
30
35
ρp/ρg
0.85
Fig. 6. Relationship between CD and ρp/ρg.
-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
φ 0.91
CD=0.855+4.8×10-4ρp/ρg
Fig. 6 showed the relationship between mass flow coefficient CD and liquid-vapor density ratio ρp/ρg at dp = 10 ȝm. The results illustrated CD will increase with density ratio, and the greater the mass loading ratio.
P = 5.5 MPa
0.90
0.89
0.89 CD 0.88
0.88
Experiment CFD
0.87
φ = 0.1 φ = 0.2 φ = 0.3
0.86
CD
CD= 0.86+82.6dp/D
0.87
CD= 0.858+62.8dp/D
0.85 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
φ Fig. 4. Comparison chart of CD - φ between experimental and simulation.
V.
0.86
0.85 0.5
FLOW CHARACTERISTICS OF V-CONE IN WET STEAM
A. Parameter Characteristics of V-Cone Firstly, the influence of vapor Reynolds number Resg and potential to kinetic energy ratio P ρ gU sg2 was considered. The results showed that CD not related Resg and P ρ gU sg2 . The maximum error didn’t exceed 0.2%. Hence Resg can be set as a constant, 1h105 which is a typical experimental point. CD= 0.853+0.075φ CD= 0.853+0.07φ
0.88 CD
ρp/ρg= 33.8 ρp/ρg= 27.3
0.86
ρp/ρg= 22.25
0.85 -0.1
0.0
0.1
0.2
0.3
φ
1.5
dp/D
2.0
2.5
−4 3.0 (×10 )
Fig. 7. Relationship between CD and dp/D.
Lastly, the relationship between mass flow coefficient CD and dimensionless droplet diameter dp/D at P = 4.6 MPa was shown in Fig. 7. It indicated that CD increases with dp/D. Besides, Fig. 6 and Fig. 7 also showed the values of CD increase with φ. B. Comparative Analysis
TABLE III.
CD= 0.853+0.065φ
0.87
1.0
Then, the effects of mass loading ratio φ, liquid-vapor density ratio ρp/ρg and dimensionless droplet diameter dp/D on mass flow coefficient CD were summarized in TABLE III. .
0.90 0.89
CD= 0.854+46.5dp/D
0.4
0.5
0.6
0.7
0.8
Fig. 5. Relationship between CD and φ.
Fig. 5 showed the relationship between mass flow coefficient CD and mass loading ratio φ at P = 4.6 MPa and dp = 10 ȝm. It found that CD significantly increases with φ, which indicates it would cause great error if we treat wet steam as single-phase flow. In addition, Fig. 5 also showed the greater the density ratio is, the larger the growth rate of CD cause, implying liquid-vapor density ratio has little effect on CD.
COMPARISON OF DIMENSIONLESS PARAMETERS’ EFFECT
Dimensionless parameter
Sign
Mass loading ratio Liquid-vapor density ratio Dimensionless droplet diameter
ȡp/ȡg d p /D
Range of Variable 0~0.7 15.5~34 (8~28)×105
φ
Relative rate of change 5.3%~5.6% 1%~1.1% 1.1%~1.9%
CD changes dramatically with mass loading ratio φ and the influence of liquid-vapor density ratio and dimensionless droplet diameter is relatively small. In wet steam, the flow characteristics of V-Cone is mainly determined by the dominate drag force f, which written as G
f =
G
ρ g Cd u g − u p dp 24vg ρp
(14)
It is obvious that liquid-vapor density ratio ρp/ρg and dimensionless droplet diameter dp/D would directly influence the drag force f; besides, the increasing ρp/ρg and dp would lead to the enhancement of interphase velocity slip. These two aspects would induce the change of interphase interaction and the flow field; eventually cause the change of CD. The effect of mass loading ratio φ is mainly reflected in the number of droplets, which also lead to the changing of interphase velocity slip, and further the drag force and CD. In addition, the droplet diameter is coupled together with pressure, temperature and mass loading ratio etc. in actual wet steam, rather than an independent variable. According to the measurement results of Yuan et al. [15] when temperature was 371.3 K, the droplet diameter is mainly between 3~5 ȝm in wet steam, which indicates that droplet diameter will not have an obvious change in actual flow, and the effect of dimensionless droplet diameter is not obvious. Consequently, the flow characteristics of V-Cone are mainly affected by liquid-vapor density ratio and mass loading ratio. VI. CONCLUSIONS Five dimensionless parameters were proposed through dimensional analysis to study mass flow coefficient CD of V-Cone flowmeter in wet steam. CFD simulation was carried out to analyze the effects of these parameters. The simulation results coincide well with the experimental results, and the detailed conclusions about these factors are as follows: 1) The flow characteristic of V-Cone is only related with mass loading ratio φ, liquid-vapor density ratio ρp/ρg and dimensionless droplet diameter dp/D. Besides, the effect of vapor Reynolds number Resg and potential to kinetic energy ratio P ρ gU sg2 is very small and could be ignored. 2) The mass flow coefficient CD is in positive correlation with mass loading ratio φ, liquid-vapor density ratio ρp/ρg and dimensionless droplet diameter dp/D, where φ is the main factor. 3) The effects of φ, ρp/ρg and dp/D on CD were explained by the changing of interphase velocity slip and dominate drag force f, which will affect the change of the flow field, and then the CD. ACKNOWLEDGMENT
This work is supported by National Natural Science Foundation of China under Grant 61627803, 51506148 and 61673291, Natural Science Foundation of Tianjin under Grant 16JCQNJC03700 and 15JCYBJC19200. REFERENCES [1] [2] [3]
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