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Modelling the growth of large rime ice accretions
Accepted Manuscript Modelling the growth of large rime ice accretions
Lasse Makkonen, Jian Zhang, Timo Karlsson, Mikko Tiihonen PII: DOI: Reference:
S0165-232X(17)30491-3 doi:10.1016/j.coldregions.2018.03.014 COLTEC 2554
To appear in:
Cold Regions Science and Technology
Received date: Revised date: Accepted date:
17 October 2017 15 March 2018 15 March 2018
Please cite this article as: Lasse Makkonen, Jian Zhang, Timo Karlsson, Mikko Tiihonen , Modelling the growth of large rime ice accretions. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Coltec(2018), doi:10.1016/j.coldregions.2018.03.014
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Modelling the growth of large rime ice accretions Lasse Makkonena* , Jian Zhanga,b, Timo Karlssona, Mikko Tiihonena a
VTT Technical Research Centre of Finland, Espoo, Finland North China Electric Power University, Beijing, China
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* Corresponding author, E-mail: [email protected]
ACCEPTED MANUSCRIPT Abstract
The conventional theory of droplet collision with an object can be used only down to a collision efficiency of about 0.1. Therefore, no accurate modeling of in-cloud icing has been possible when droplets are small, wind speed is low, or the object is large. This has also put a limit on the size of an ice accretion on e.g. a power line cable up to which its growth can be simulated. We utilize
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results of icing wind tunnel experiments and fluid dynamics simulations to explain the differences
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between the experiments and the theory when the collision efficiency is small. We confirm that the history term in the droplet trajectory equations becomes relevant at small collision efficiencies.
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Including this term, and applying the integral over the size distribution instead of using the median
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volume diameter, shows that accurate modeling of icing at very small collision efficiencies is
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feasible. This makes it possible to estimate large rime ice loads relevant in structural design.
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Keywords—ice accretion; rime ice; collision efficiency; history term; droplet distribution; icing experiments
ACCEPTED MANUSCRIPT 1.
Introduction
When rime icing occurs, the growth mode is “dry”, i.e., all droplets that impinge on the object contribute to the ice load formed. Under these conditions, the rate of icing is proportional to the droplet collision efficiency E. Specifically, the rate of increase of the ice load dM/dt, is then
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(1)
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dM EUWDL dt
where E is the mean volumetric collision efficiency, U is the relative velocity of the free two-phase
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flow with respect to the object, W is liquid water content in the flow, D is the cross-sectional width
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of the object, and L is its length in perpendicular to the air flow.
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In this paper, our main interest is a stationary cylindrical object, such as a power line cable, or a tubular component of a tower. We may then take U as the wind speed, D as the cylinder diameter, and express the icing rate per unit length as R = (dM/dt)/L. We also assume that D/L is sufficiently
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small so that the process is two-dimensional. With these assumptions in mind, equation (1) can also
E
R UWD
(2)
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be understood as the definition of the collision efficiency:
When a droplet suspended in the air flow approaches a cylinder, its inertia tends to keep the trajectory straight. However, the air must flow around the object, and a droplet tends to follow the streamlines due to the friction between the droplet and air. Thus, the trajectory of a droplet around a cylinder is determined by the balance between inertial and frictional forces (Albrecht, 1931). The overall collision efficiency E is, therefore, determined by such a balance, considering the integral over all droplets that are initially aimed at the cross-section of the cylinder. Moreover, because a cloud consists of a distribution of droplets, the integral must include droplets of all sizes.
ACCEPTED MANUSCRIPT Thus, even in the simple 2-D case of a cylinder in a laminar flow, the physical process that determines the overall collision efficiency E is complex. Historically, this complexity prevented the theoretical assessment of E until numerical calculation of droplet trajectories became possible. Immediately after the first computer was available, it was used to solve this particular problem. Langmuir and Blodgett (1946) then solved the droplet trajectory equations in a potential flow and
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determined theoretically E for various parameter combinations. Langmuir and Blodgett (1946) showed that, by assuming a reasonable parameterization for droplet-air friction, E can be calculated
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by two dimensionless parameters, Stokes number K
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w d 2U 9 D
(3)
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K
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and Langmuir parameter Φ
= Re2 / K
(4)
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where Re = Udρa/μ is the droplet Reynolds number. Here d is the droplet diameter, ρw the density of water and μ the dynamic viscosity of air. The theory thus allows scaling of a rime ice accretion
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process (Makkonen and Oleskiw, 1997).
Langmuir and Blodgett (1946) presented their results in the form of graphs and fitted equations.
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Other mathematical formulations of their data have subsequently been proposed by e.g. Makkonen (1984) and Finstad et al. (1988a). Simulations of E have also been made by the finite element method (McComber and Touzot, 1981). More recently, the collision efficiency has been determined by simulating droplet trajectories using computational fluid dynamics software (Nakakita et al., 2010; Zhang et al. 2017).
Langmuir and Blodgett (1946) avoided the complexity arising from the variety of droplet sizes by using a single parameter, the median volume droplet diameter (MVD), as representative of the droplet distribution. This approach was subsequently followed by other researchers albeit without a
ACCEPTED MANUSCRIPT proper basis. Eventually, Finstad et al. (1988b) showed, both by analysing measured droplet size spectra, and analytically, that MVD is indeed the optimum single distribution parameter for the calculation of E.
The theoretical predictions of the collision efficiency on circular cylinders were verified by the icing wind tunnel experiments by Makkonen and Stallabrass (1987). They showed that when E >
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0.07, the theory fully agrees with experimental data considering the accuracy by which the input
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parameters can be measured. This accuracy is limited due to the sensitivity of E to small changes in the MVD. On the other hand, this sensitivity can be used to accurately determine the MVD by
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measuring ice amounts simultaneously on cylinders of various diameters (Howe, 1991; Makkonen,
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1992; Knetzevici et al., 2005).
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It has been observed (Makkonen, 1992) that this, so called rotating multi-cylinder method, is poorly applicable at small wind speeds. Moreover, it has been pointed out that the trajectory equations may be inaccurate when K is small, due to a “history term” not included in the theoretical
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trajectory calculations (Temkin and Mehta, 1982; Oleskiw, 1982; Finstad et al. 1988a). Strictly
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speaking, this means that the theory used in calculating E is invalid when E is very small. Moreover, it has been suggested that the roughness of a cylinder surface due to icing affects the
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airflow and droplet trajectories when E is small (List, 1977; Personne and Gayet, 1988). The error caused by taking MVD as the single representative parameter of the droplet distribution, instead of integrating E over the whole size distribution, is also likely bigger when E is very small. Considering all these aspects, it is noteworthy that the successful experimental verification of the collision efficiency theory extends down to only E = 0.07 (Makkonen and Stallabrass, 1987).
Consequently, when the collision efficiency E is smaller than 0.07, there has been no icing model that would be based on a well-founded and verified theory. This has generally not been seen as a problem, because in case of moving objects, such as aircraft and wind turbine blades, the collision efficiency is typically high due to the high flow speed. For power lines and tower
ACCEPTED MANUSCRIPT structures, the main application of icing modelling is to assess the extreme ice loads for engineering design, which are also often formed at high wind speeds.
However, design ice loads may grow also by cumulative icing over a long period. When the duration of the event is long, the ice accretion may grow large even at low wind speeds. This is illustrated in Figs. 1 and 2. The diameter of the ice accretions on the power line cables shown in
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Fig. 1 is one metre, and that of the cylindrical structure shown in Fig. 2 even more. For cylinders
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this large, the conventional theory of the collision efficiency, discussed above, predicts E = 0 for any combination of V and MVD that may occur in atmospheric icing conditions. The question then
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is: How is it possible that these ice accretions have grown so large? Considering that the ice loads
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in Figs. 1 and 2 represent extreme events that are of interest in structural design, this question is
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certainly a relevant one.
Fig. 1. Rime ice on power lines in Norway (photo: Olav Wist)
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Fig. 2. Rime ice on a cylindrical building
The discussion above suggests that the ice accretion process on a very large object is poorly
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understood, and that present icing models thus provide no reliable means for modelling long-term
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large rime ice loads. Furthermore, no systematic experimental data exist on icing at a very low E. In this paper, we present results from icing wind tunnel experiments of E on large cylinders at low wind speeds, and interpret them by comparisons with simulation results obtained by computational
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fluid dynamics software. We thus develop a tool for estimating ice loads also when E is very small.
2.
Experiments
The experiments were conducted in the VTT icing wind tunnel. The tunnel is an “open-loop” tunnel placed entirely inside a large cold room. The cross-section of the tunnel mouth is 0.7m*0.7m. Ice was grown on 0.157m long smooth aluminium cylinders, 0.03-0.17m in diameter. The cylinders were placed vertically and rotated by a motor at 5 rpm. To rule out the effect of
ACCEPTED MANUSCRIPT blockage, the cylinders were placed in front of the mouth of the tunnel. To ensure uniform ice growth and minimize border effects, the cylinders were equipped with two thin metal sheets at both
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ends, as shown in Fig. 3.
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Fig. 3. Test cylinder after an icing experiment
The temperature and wind speed in the test section were measured using calibrated sensors. The liquid water content (LWC) was calibrated for each wind speed and temperature pair by measuring the ice growth on a 30mm cylinder and using the formulas defined in ISO 12494 (ISO, 2001). Droplet cloud uniformity was verified by icing of a mesh placed across IWT test section. The results showed that uniformity was good at the central part of the test section, where the test cylinders were placed during the test runs. Under the test conditions, LWC was 0.4 g/m3 . Air temperature was -5 o C and two wind speeds 4 m/s and 7 m/s were used. The droplet size distribution in the tunnel has been calibrated earlier using The Cloud, Aerosol and Precipitation
ACCEPTED MANUSCRIPT Spectrometer probe (CAPS, Droplet Measurement Technologies, Boulder, CO, USA). This droplet size distribution is shown in Fig. 4. The median volume diameter (MVD) of this distribution, calculated following Finstad et al. (1988b), is 16 μm.
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MVD D
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Fig. 4. Droplet size distribution in the icing wind tunnel.
Fig. 5. Growth of ice on the surface of the 100 mm cylinder. The photos are from the same experiment, but not exactly from the same spot on the surface.
ACCEPTED MANUSCRIPT The test cylinders were weighted at the beginning of a test and then every 30 minutes. For measuring the differential ice weight accurately, the heavier cylinders were iced up for a longer time, up to three hours.
In the experiments, ice started appearing on the cylinder as discrete spots around the cylinder
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surface. As the experiment went on, ice grew mainly on these existing ice spots, resulting in granular and soft rime on the cylinder. A typical surface structure of ice in the experiments is
Simulations
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3.
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shown in Fig. 5.
In natural conditions, the volume fraction of water is small, so that the Euler-Lagrangian model can be used for modelling. In the model, air is treated as continuum by solving the Navier-Stokes
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equations, while water droplet motion is solved by tracking them through the calculated flow field.
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Considering the symmetry, the flow is considered to be a 2D-flow, where the x-axis and y-axis are
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defined in Fig. 6. The airflow is considered to be incompressible and steady.
Fig. 6. Sketch of the flow in the model.
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The governing equation for air is as follows (Fluent Inc., 2006a).
(1) Mass conservation equation
uy y
where ux is the x-axial velocity, and uy is y-axial velocity.
uy x
uy
uy y
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ux ux 2 ux 2 ux p uy ) ( 2 2 ) x y x x y )
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a (ux (u a x
(5)
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(2) Momentum conservation equation
0
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x
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ux
2uy 2uy p ( 2 2 ) y x y
(6)
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where p is the static air pressure, μ is the dynamic viscosity of air.
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(3) Turbulence model
The 2D Spalart-Allmaras model is selected here as the turbulence model, because this model is quite suitable for simulating the airflow during ice accretion on conductors, wind turbines and
a kux x
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aircraft (Guo et al., 2010). This is a one-equation model, which is easy to solve and accurate. a kuy y
2 2 t k t k k 1 k C C Gμ Yk b2 a b2 a k x k x y k y x y
(7) Here, k is the molecular kinematic viscosity, and Gμ is the production of turbulent energy, and Yk is the dissipation of turbulent energy that occurs in the near-wall region due to wall blocking and viscous damping. σk and Cb2 are the constants, which are 1.5 and 0.622 respectively (Fluent Inc., 2006a).
ACCEPTED MANUSCRIPT The equation of motion for a water droplet is g w a du w fd u uw F dt w
(8)
where uw is the droplet velocity, f d(u-uw) is the drag force per unit water mass, and g(ρw-ρa)/ρw is resultant of gravity and buoyancy per unit water mass. In Eq. (8), F is a non-steady state drag term,
0.5
t
du w d
d
t
(9)
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F 2p w d πw
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18 w
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which is also known as the history term. F is given as (Landau and Lifshitz, 1959)
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The history term describes the influence of vorticity diffusion from the accelerating droplet surface. The vorticity induced in the airflow depends on the accelerating motion of the water drop.
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The actual drag experienced at any time is affected more by the recent history than the history term. At small collision efficiencies, the history term will decrease the droplet’s deceleration and increase
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both its velocity near the surface and the collision efficiency.
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To reduce the impact of boundaries on the flow, the calculation mesh is constructed so that the length of each boundary is 30 times the diameter of the cylinder. Gambit© is used for modelling,
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where structural quadrilateral grid partition is used for meshing (Fluent Inc., 2006b).
In this model, the velocity boundary condition, which defines the velocity of the inlet, is chosen for the inlet. The outflow boundary condition, which defines a zero diffusion flux for all flow variables, is chosen for the outlet. The no-slip solid wall boundary condition, which means that the velocity on the wall is zero, is applied. The two-dimensional uncoupled implicit solver based on the pressure method is used as the solver, the Spalart-Allmaras model is selected as the turbulence model, and the wall is treated as the standard wall function (Fluent Inc., 2006a). The SIMPLE algorithm in Fluent© is used to solve the discrete equations, and the discrete scheme is “secondorder upwind”. All the selections of Fluent© option switches are the standard ones.
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4. Results and discussion
Figure 7 shows the ice mass measured in the experiments on the five cylinders as a function of time. These result show that, in the case of a low wind speed, the combined effect of the decreasing
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collision efficiency and increasing collecting object area is such that the accreted ice mass is almost independent of the initial cylinder diameter. This agrees with the estimated effect of a conductor
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diameter on the ice load under these conditions (Makkonen, 1986) and supports the approach of the
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ISO 12494 standard (ISO, 2001).
The results of the collision efficiency from the experiments are presented in the right-hand
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column of Table 1, marked by bold numbers. This experimental E was calculated from Eq. (2) at
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the end of an experiment.
Fig. 7. Time evolution of the ice mass in the icing wind tunnel experiments at U=4m/s.
Table 1 shows, for comparison, values of three different collision efficiencies calculated by the simulation model. The first one (left) is E calculated using the MVD and including the history term. The second one (centre) is E calculated by separately simulating the collision efficiency for each
ACCEPTED MANUSCRIPT bin of the droplet size distribution and neglecting the history term. The third simulated collision efficiency (right) is calculated by separately simulating E for each bin of the droplet size distribution and including the history term. The agreement between the theory and the experiments is illustrated in Fig. 8. Table 1 and Fig. 8 show that, at small E, employing the whole distribution in the simulation is much more accurate
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than using the single distribution parameter, MVD.
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Table 1. Results of the simulations and icing wind tunnel experiments.
Simulated E based on the distribution and with the history term
0.092
0.092
0.085
0.165
0.165
0.177
0.039
0.039
0.031
0.017
0.022
0.020
0
0.011
0.017
0.017
0.008
0.028
0.038
0.036
0
0.005
0.011
0.010
Cylinder diameter (m)
4
0.03
0.043
7
0.03
0.115
4
0.05
0
4
0.08
0
4
0.1
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0.1
4
0.17
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Wind velocity (m/s)
Simulated E based on MVD and with the history term
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Simulated E based on the distribution and without the history term
Experimental E
This is mainly because calculation by the MVD results in E(MVD) = 0 at a very small MVD, while the biggest droplets of the same droplet size distribution may involve E > 0. This situation is illustrated by two examples in Fig. 9 based on simulations that include the history term.
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Fig. 8. Theoretical collision efficiency vs. experimental collision efficiency for the three
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different simulations.
Fig. 9. Simulated collision efficiency as a function of the droplet diameter at U = 4m/s, D=0.1m (solid line) and at U = 20m/s, D=0.5m (dashed line).
Table 1 and Fig. 8 also show that including the history term, given in Eq. (9), clearly improves the simulation results at a very small E. For the smallest cylinder, the collision efficiency with and
ACCEPTED MANUSCRIPT without history term is essentially the same, since K > 0.25. However, for the other cases, E simulated including the history term is in good agreement with the experimental values, while E simulated without the history term includes large errors.
Most importantly, Table 1 and Fig. 8 show that there is an excellent agreement between the simulated and measured E when the entire droplet size distribution is employed and the history
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term is included. Based on these results, we may now extend the theoretical assessment of ice loads
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from its previous validity limit of E=0.07 down to E=0.01. This provides a useful tool for
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estimating large cumulative ice loads, and ice formed on large objects and at low wind speeds.
This study was concerned with the fundamental case of a smooth circular cylinder. To that end,
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our icing experiments were made on slowly rotating cylinders. These data may be applied to e.g.
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estimating icing on overhead power line cables of different diameters (Makkonen, 1986). Moreover, the typical rime ice shape on a power line is cylindrical, but the diameter changes with time (Makkonen, 1984). Simulation of such a process up to a very large accretion is now feasible
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based on the results of this study. In the future, comparisons of icing experiments and fluid
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dynamics simulations should be made for other shapes and for irregular surfaces. This will allow more accurate estimation of ice loads on structural components and simulation of evolving complex
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ice shapes.
Acknowledegements
This work was supported by the Nordic project Frontlines No. 245370. The Fund from China Scholarship Council and support by the Fundamental Research Funds for the Central Universities of China (2015XS93) is greatly acknowledged by the second author.
ACCEPTED MANUSCRIPT References
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Finstad, K.F., Lozowski, E.P., Gates, E.M., 1988a. A computational investigation of water droplet
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trajectories. J. Atmos. Oceanic Technol. 5, 160-170.
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Finstad, K.J., Lozowski, E.P., Makkonen, L., 1988b. On the median volume diameter
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approximation for droplet collision efficiency. J. Atmos. Sci. 45, 4008-4012.
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Fluent Inc., 2006b. Gambit user’s guide.
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Fluent Inc., 2006a. Fluent user’s guide.
Guo, H., Heyun, Liu, Xiaosong, Gu, 2010. Numerical simulation of the local collection efficiency on icing conductor. In: International Conference on Electrical and Control Engineering, pp. 2868–
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2871.
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Howe, J., 1991. Rotating Multicylinder Method for the Measurement of Cloud Liquid-Water Content and Droplet Size. U.S Army Cold Regions Research & Engineering Laboratory CRREL
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Report 91-2, Hanover, NH, 21 pp.
International Standardization Organization (ISO), 2000. International Standard for Atmospheric Icing of Structures, ISO 12494.
Knetzevici, D., Kind, R. J. Oleskiw, M.M., 2005. Determination of medium volume diameter (MVD) and liquid water content (LWC) by multiple rotating c ylinders. 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, U.S.A. doi.org/10.2514/6.2005-861.
Landau, L., Lifshitz, E.M., 1959. Fluid Mechanics. Pergamon Press, pp. 96-98.
ACCEPTED MANUSCRIPT Langmuir, I., Blodgett, K., 1946. A mathematical investigation of water droplet trajectories. Collected Works of I. Langmuir, vol. 10, Pergamon Press, 348–393.
List, R., 1977. Ice accretion on structures. J. Glaciol. 19, 451-466.
Makkonen, L., 1984. Modeling of ice accretion on wires. J. Clim. Appl. Meteorol. 23, 929-939.
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Makkonen, L, 1986. The effect of conductor diameter on ice load as determined by a numerical
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icing model. Proceedings, Third International Workshop on Atmospheric Icing of Structures
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(IWAIS), 81-89.
Makkonen, L., 1992. Analysis of rotating multicylinder data in measuring cloud droplet size and
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liquid water content. J. Atmos. Oceanic Technol. 9, 258-263.
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Makkonen, L., Oleskiw, M.M., 1997. Small- scale experiments on rime icing. Cold Regions Sci. Technol. 25, 173 - 182.
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Makkonen, L., Stallabrass, J.R., 1987. Experiments on the cloud droplet collision efficiency of
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cylinders. J. Clim. Appl. Meteorol. 26, 1406-1411.
McComber, P., Touzot, G., 1981. Calculation of the impingement of cloud droplets in a cylinder
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by the Finite-Element Method. J. Atmos. Sci. 38, 1027-1036.
Nakakita, K., Nadarajah, S., Habashi, W.G., 2010. Toward real-time aero-icing simulation for complete aircraft configurations via FENSAP-ICE. J. Aircraft 27, 96-109.
Oleskiw, M.M., 1982. Computer Simulation of Time-Dependent Rime Icing on Airfoils. Ph.D. thesis. University of Alberta, Canada. doi:10.7939/R3QR4NW8V.
Personne, P., Gayet, J.-F., 1988. Ice accretion on wires and anti- icing induced by Joule effect. J. Clim. Appl. Meteorol. 27, 101-114.
ACCEPTED MANUSCRIPT Temkin, S., Mehta, H.K., 1982. Droplet drag in an accelerating and decelerating flow. J. Fluid. Mech. 116, 297-313.
Zhang, J., Makkonen, L., He, Q., 2017. A 2-D numerical study on the effect of condutor shape on
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icing collision efficiency. Cold Regions Sci. Technol. 143, 52-58.
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Highlights
Fluid dynamics simulations on the droplet collision efficiency on cylinders were made at small collision efficiencies.
Icing wind tunnel experiments were made at small collision efficiencies to compare with the
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model results. It is confirmed that the history term in the droplet trajectory equations becomes relevant at
The model and experiments agree when the history term is included and the integral over
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small collision efficiencies.
the droplet size distribution used.
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Using this method, it becomes possible to estimate large cumulative rime ice accretions
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relevant in structural design.
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Lasse Makkonen, Jian Zhang, Timo Karlsson, Mikko Tiihonen PII: DOI: Reference:
S0165-232X(17)30491-3 doi:10.1016/j.coldregions.2018.03.014 COLTEC 2554
To appear in:
Cold Regions Science and Technology
Received date: Revised date: Accepted date:
17 October 2017 15 March 2018 15 March 2018
Please cite this article as: Lasse Makkonen, Jian Zhang, Timo Karlsson, Mikko Tiihonen , Modelling the growth of large rime ice accretions. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Coltec(2018), doi:10.1016/j.coldregions.2018.03.014
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Modelling the growth of large rime ice accretions Lasse Makkonena* , Jian Zhanga,b, Timo Karlssona, Mikko Tiihonena a
VTT Technical Research Centre of Finland, Espoo, Finland North China Electric Power University, Beijing, China
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* Corresponding author, E-mail: [email protected]
ACCEPTED MANUSCRIPT Abstract
The conventional theory of droplet collision with an object can be used only down to a collision efficiency of about 0.1. Therefore, no accurate modeling of in-cloud icing has been possible when droplets are small, wind speed is low, or the object is large. This has also put a limit on the size of an ice accretion on e.g. a power line cable up to which its growth can be simulated. We utilize
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results of icing wind tunnel experiments and fluid dynamics simulations to explain the differences
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between the experiments and the theory when the collision efficiency is small. We confirm that the history term in the droplet trajectory equations becomes relevant at small collision efficiencies.
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Including this term, and applying the integral over the size distribution instead of using the median
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volume diameter, shows that accurate modeling of icing at very small collision efficiencies is
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feasible. This makes it possible to estimate large rime ice loads relevant in structural design.
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Keywords—ice accretion; rime ice; collision efficiency; history term; droplet distribution; icing experiments
ACCEPTED MANUSCRIPT 1.
Introduction
When rime icing occurs, the growth mode is “dry”, i.e., all droplets that impinge on the object contribute to the ice load formed. Under these conditions, the rate of icing is proportional to the droplet collision efficiency E. Specifically, the rate of increase of the ice load dM/dt, is then
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(1)
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dM EUWDL dt
where E is the mean volumetric collision efficiency, U is the relative velocity of the free two-phase
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flow with respect to the object, W is liquid water content in the flow, D is the cross-sectional width
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of the object, and L is its length in perpendicular to the air flow.
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In this paper, our main interest is a stationary cylindrical object, such as a power line cable, or a tubular component of a tower. We may then take U as the wind speed, D as the cylinder diameter, and express the icing rate per unit length as R = (dM/dt)/L. We also assume that D/L is sufficiently
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small so that the process is two-dimensional. With these assumptions in mind, equation (1) can also
E
R UWD
(2)
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be understood as the definition of the collision efficiency:
When a droplet suspended in the air flow approaches a cylinder, its inertia tends to keep the trajectory straight. However, the air must flow around the object, and a droplet tends to follow the streamlines due to the friction between the droplet and air. Thus, the trajectory of a droplet around a cylinder is determined by the balance between inertial and frictional forces (Albrecht, 1931). The overall collision efficiency E is, therefore, determined by such a balance, considering the integral over all droplets that are initially aimed at the cross-section of the cylinder. Moreover, because a cloud consists of a distribution of droplets, the integral must include droplets of all sizes.
ACCEPTED MANUSCRIPT Thus, even in the simple 2-D case of a cylinder in a laminar flow, the physical process that determines the overall collision efficiency E is complex. Historically, this complexity prevented the theoretical assessment of E until numerical calculation of droplet trajectories became possible. Immediately after the first computer was available, it was used to solve this particular problem. Langmuir and Blodgett (1946) then solved the droplet trajectory equations in a potential flow and
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determined theoretically E for various parameter combinations. Langmuir and Blodgett (1946) showed that, by assuming a reasonable parameterization for droplet-air friction, E can be calculated
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by two dimensionless parameters, Stokes number K
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w d 2U 9 D
(3)
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K
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and Langmuir parameter Φ
= Re2 / K
(4)
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where Re = Udρa/μ is the droplet Reynolds number. Here d is the droplet diameter, ρw the density of water and μ the dynamic viscosity of air. The theory thus allows scaling of a rime ice accretion
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process (Makkonen and Oleskiw, 1997).
Langmuir and Blodgett (1946) presented their results in the form of graphs and fitted equations.
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Other mathematical formulations of their data have subsequently been proposed by e.g. Makkonen (1984) and Finstad et al. (1988a). Simulations of E have also been made by the finite element method (McComber and Touzot, 1981). More recently, the collision efficiency has been determined by simulating droplet trajectories using computational fluid dynamics software (Nakakita et al., 2010; Zhang et al. 2017).
Langmuir and Blodgett (1946) avoided the complexity arising from the variety of droplet sizes by using a single parameter, the median volume droplet diameter (MVD), as representative of the droplet distribution. This approach was subsequently followed by other researchers albeit without a
ACCEPTED MANUSCRIPT proper basis. Eventually, Finstad et al. (1988b) showed, both by analysing measured droplet size spectra, and analytically, that MVD is indeed the optimum single distribution parameter for the calculation of E.
The theoretical predictions of the collision efficiency on circular cylinders were verified by the icing wind tunnel experiments by Makkonen and Stallabrass (1987). They showed that when E >
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0.07, the theory fully agrees with experimental data considering the accuracy by which the input
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parameters can be measured. This accuracy is limited due to the sensitivity of E to small changes in the MVD. On the other hand, this sensitivity can be used to accurately determine the MVD by
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measuring ice amounts simultaneously on cylinders of various diameters (Howe, 1991; Makkonen,
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1992; Knetzevici et al., 2005).
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It has been observed (Makkonen, 1992) that this, so called rotating multi-cylinder method, is poorly applicable at small wind speeds. Moreover, it has been pointed out that the trajectory equations may be inaccurate when K is small, due to a “history term” not included in the theoretical
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trajectory calculations (Temkin and Mehta, 1982; Oleskiw, 1982; Finstad et al. 1988a). Strictly
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speaking, this means that the theory used in calculating E is invalid when E is very small. Moreover, it has been suggested that the roughness of a cylinder surface due to icing affects the
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airflow and droplet trajectories when E is small (List, 1977; Personne and Gayet, 1988). The error caused by taking MVD as the single representative parameter of the droplet distribution, instead of integrating E over the whole size distribution, is also likely bigger when E is very small. Considering all these aspects, it is noteworthy that the successful experimental verification of the collision efficiency theory extends down to only E = 0.07 (Makkonen and Stallabrass, 1987).
Consequently, when the collision efficiency E is smaller than 0.07, there has been no icing model that would be based on a well-founded and verified theory. This has generally not been seen as a problem, because in case of moving objects, such as aircraft and wind turbine blades, the collision efficiency is typically high due to the high flow speed. For power lines and tower
ACCEPTED MANUSCRIPT structures, the main application of icing modelling is to assess the extreme ice loads for engineering design, which are also often formed at high wind speeds.
However, design ice loads may grow also by cumulative icing over a long period. When the duration of the event is long, the ice accretion may grow large even at low wind speeds. This is illustrated in Figs. 1 and 2. The diameter of the ice accretions on the power line cables shown in
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Fig. 1 is one metre, and that of the cylindrical structure shown in Fig. 2 even more. For cylinders
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this large, the conventional theory of the collision efficiency, discussed above, predicts E = 0 for any combination of V and MVD that may occur in atmospheric icing conditions. The question then
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is: How is it possible that these ice accretions have grown so large? Considering that the ice loads
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in Figs. 1 and 2 represent extreme events that are of interest in structural design, this question is
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certainly a relevant one.
Fig. 1. Rime ice on power lines in Norway (photo: Olav Wist)
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Fig. 2. Rime ice on a cylindrical building
The discussion above suggests that the ice accretion process on a very large object is poorly
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understood, and that present icing models thus provide no reliable means for modelling long-term
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large rime ice loads. Furthermore, no systematic experimental data exist on icing at a very low E. In this paper, we present results from icing wind tunnel experiments of E on large cylinders at low wind speeds, and interpret them by comparisons with simulation results obtained by computational
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fluid dynamics software. We thus develop a tool for estimating ice loads also when E is very small.
2.
Experiments
The experiments were conducted in the VTT icing wind tunnel. The tunnel is an “open-loop” tunnel placed entirely inside a large cold room. The cross-section of the tunnel mouth is 0.7m*0.7m. Ice was grown on 0.157m long smooth aluminium cylinders, 0.03-0.17m in diameter. The cylinders were placed vertically and rotated by a motor at 5 rpm. To rule out the effect of
ACCEPTED MANUSCRIPT blockage, the cylinders were placed in front of the mouth of the tunnel. To ensure uniform ice growth and minimize border effects, the cylinders were equipped with two thin metal sheets at both
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ends, as shown in Fig. 3.
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Fig. 3. Test cylinder after an icing experiment
The temperature and wind speed in the test section were measured using calibrated sensors. The liquid water content (LWC) was calibrated for each wind speed and temperature pair by measuring the ice growth on a 30mm cylinder and using the formulas defined in ISO 12494 (ISO, 2001). Droplet cloud uniformity was verified by icing of a mesh placed across IWT test section. The results showed that uniformity was good at the central part of the test section, where the test cylinders were placed during the test runs. Under the test conditions, LWC was 0.4 g/m3 . Air temperature was -5 o C and two wind speeds 4 m/s and 7 m/s were used. The droplet size distribution in the tunnel has been calibrated earlier using The Cloud, Aerosol and Precipitation
ACCEPTED MANUSCRIPT Spectrometer probe (CAPS, Droplet Measurement Technologies, Boulder, CO, USA). This droplet size distribution is shown in Fig. 4. The median volume diameter (MVD) of this distribution, calculated following Finstad et al. (1988b), is 16 μm.
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MVD D
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Fig. 4. Droplet size distribution in the icing wind tunnel.
Fig. 5. Growth of ice on the surface of the 100 mm cylinder. The photos are from the same experiment, but not exactly from the same spot on the surface.
ACCEPTED MANUSCRIPT The test cylinders were weighted at the beginning of a test and then every 30 minutes. For measuring the differential ice weight accurately, the heavier cylinders were iced up for a longer time, up to three hours.
In the experiments, ice started appearing on the cylinder as discrete spots around the cylinder
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surface. As the experiment went on, ice grew mainly on these existing ice spots, resulting in granular and soft rime on the cylinder. A typical surface structure of ice in the experiments is
Simulations
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3.
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shown in Fig. 5.
In natural conditions, the volume fraction of water is small, so that the Euler-Lagrangian model can be used for modelling. In the model, air is treated as continuum by solving the Navier-Stokes
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equations, while water droplet motion is solved by tracking them through the calculated flow field.
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Considering the symmetry, the flow is considered to be a 2D-flow, where the x-axis and y-axis are
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defined in Fig. 6. The airflow is considered to be incompressible and steady.
Fig. 6. Sketch of the flow in the model.
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The governing equation for air is as follows (Fluent Inc., 2006a).
(1) Mass conservation equation
uy y
where ux is the x-axial velocity, and uy is y-axial velocity.
uy x
uy
uy y
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ux ux 2 ux 2 ux p uy ) ( 2 2 ) x y x x y )
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a (ux (u a x
(5)
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(2) Momentum conservation equation
0
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x
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ux
2uy 2uy p ( 2 2 ) y x y
(6)
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where p is the static air pressure, μ is the dynamic viscosity of air.
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(3) Turbulence model
The 2D Spalart-Allmaras model is selected here as the turbulence model, because this model is quite suitable for simulating the airflow during ice accretion on conductors, wind turbines and
a kux x
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aircraft (Guo et al., 2010). This is a one-equation model, which is easy to solve and accurate. a kuy y
2 2 t k t k k 1 k C C Gμ Yk b2 a b2 a k x k x y k y x y
(7) Here, k is the molecular kinematic viscosity, and Gμ is the production of turbulent energy, and Yk is the dissipation of turbulent energy that occurs in the near-wall region due to wall blocking and viscous damping. σk and Cb2 are the constants, which are 1.5 and 0.622 respectively (Fluent Inc., 2006a).
ACCEPTED MANUSCRIPT The equation of motion for a water droplet is g w a du w fd u uw F dt w
(8)
where uw is the droplet velocity, f d(u-uw) is the drag force per unit water mass, and g(ρw-ρa)/ρw is resultant of gravity and buoyancy per unit water mass. In Eq. (8), F is a non-steady state drag term,
0.5
t
du w d
d
t
(9)
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F 2p w d πw
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18 w
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which is also known as the history term. F is given as (Landau and Lifshitz, 1959)
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The history term describes the influence of vorticity diffusion from the accelerating droplet surface. The vorticity induced in the airflow depends on the accelerating motion of the water drop.
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The actual drag experienced at any time is affected more by the recent history than the history term. At small collision efficiencies, the history term will decrease the droplet’s deceleration and increase
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both its velocity near the surface and the collision efficiency.
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To reduce the impact of boundaries on the flow, the calculation mesh is constructed so that the length of each boundary is 30 times the diameter of the cylinder. Gambit© is used for modelling,
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where structural quadrilateral grid partition is used for meshing (Fluent Inc., 2006b).
In this model, the velocity boundary condition, which defines the velocity of the inlet, is chosen for the inlet. The outflow boundary condition, which defines a zero diffusion flux for all flow variables, is chosen for the outlet. The no-slip solid wall boundary condition, which means that the velocity on the wall is zero, is applied. The two-dimensional uncoupled implicit solver based on the pressure method is used as the solver, the Spalart-Allmaras model is selected as the turbulence model, and the wall is treated as the standard wall function (Fluent Inc., 2006a). The SIMPLE algorithm in Fluent© is used to solve the discrete equations, and the discrete scheme is “secondorder upwind”. All the selections of Fluent© option switches are the standard ones.
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4. Results and discussion
Figure 7 shows the ice mass measured in the experiments on the five cylinders as a function of time. These result show that, in the case of a low wind speed, the combined effect of the decreasing
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collision efficiency and increasing collecting object area is such that the accreted ice mass is almost independent of the initial cylinder diameter. This agrees with the estimated effect of a conductor
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diameter on the ice load under these conditions (Makkonen, 1986) and supports the approach of the
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ISO 12494 standard (ISO, 2001).
The results of the collision efficiency from the experiments are presented in the right-hand
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column of Table 1, marked by bold numbers. This experimental E was calculated from Eq. (2) at
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the end of an experiment.
Fig. 7. Time evolution of the ice mass in the icing wind tunnel experiments at U=4m/s.
Table 1 shows, for comparison, values of three different collision efficiencies calculated by the simulation model. The first one (left) is E calculated using the MVD and including the history term. The second one (centre) is E calculated by separately simulating the collision efficiency for each
ACCEPTED MANUSCRIPT bin of the droplet size distribution and neglecting the history term. The third simulated collision efficiency (right) is calculated by separately simulating E for each bin of the droplet size distribution and including the history term. The agreement between the theory and the experiments is illustrated in Fig. 8. Table 1 and Fig. 8 show that, at small E, employing the whole distribution in the simulation is much more accurate
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than using the single distribution parameter, MVD.
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Table 1. Results of the simulations and icing wind tunnel experiments.
Simulated E based on the distribution and with the history term
0.092
0.092
0.085
0.165
0.165
0.177
0.039
0.039
0.031
0.017
0.022
0.020
0
0.011
0.017
0.017
0.008
0.028
0.038
0.036
0
0.005
0.011
0.010
Cylinder diameter (m)
4
0.03
0.043
7
0.03
0.115
4
0.05
0
4
0.08
0
4
0.1
7
0.1
4
0.17
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Wind velocity (m/s)
Simulated E based on MVD and with the history term
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Simulated E based on the distribution and without the history term
Experimental E
This is mainly because calculation by the MVD results in E(MVD) = 0 at a very small MVD, while the biggest droplets of the same droplet size distribution may involve E > 0. This situation is illustrated by two examples in Fig. 9 based on simulations that include the history term.
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Fig. 8. Theoretical collision efficiency vs. experimental collision efficiency for the three
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different simulations.
Fig. 9. Simulated collision efficiency as a function of the droplet diameter at U = 4m/s, D=0.1m (solid line) and at U = 20m/s, D=0.5m (dashed line).
Table 1 and Fig. 8 also show that including the history term, given in Eq. (9), clearly improves the simulation results at a very small E. For the smallest cylinder, the collision efficiency with and
ACCEPTED MANUSCRIPT without history term is essentially the same, since K > 0.25. However, for the other cases, E simulated including the history term is in good agreement with the experimental values, while E simulated without the history term includes large errors.
Most importantly, Table 1 and Fig. 8 show that there is an excellent agreement between the simulated and measured E when the entire droplet size distribution is employed and the history
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term is included. Based on these results, we may now extend the theoretical assessment of ice loads
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from its previous validity limit of E=0.07 down to E=0.01. This provides a useful tool for
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estimating large cumulative ice loads, and ice formed on large objects and at low wind speeds.
This study was concerned with the fundamental case of a smooth circular cylinder. To that end,
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our icing experiments were made on slowly rotating cylinders. These data may be applied to e.g.
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estimating icing on overhead power line cables of different diameters (Makkonen, 1986). Moreover, the typical rime ice shape on a power line is cylindrical, but the diameter changes with time (Makkonen, 1984). Simulation of such a process up to a very large accretion is now feasible
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based on the results of this study. In the future, comparisons of icing experiments and fluid
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dynamics simulations should be made for other shapes and for irregular surfaces. This will allow more accurate estimation of ice loads on structural components and simulation of evolving complex
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ice shapes.
Acknowledegements
This work was supported by the Nordic project Frontlines No. 245370. The Fund from China Scholarship Council and support by the Fundamental Research Funds for the Central Universities of China (2015XS93) is greatly acknowledged by the second author.
ACCEPTED MANUSCRIPT References
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trajectories. J. Atmos. Oceanic Technol. 5, 160-170.
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Finstad, K.J., Lozowski, E.P., Makkonen, L., 1988b. On the median volume diameter
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approximation for droplet collision efficiency. J. Atmos. Sci. 45, 4008-4012.
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Fluent Inc., 2006b. Gambit user’s guide.
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Fluent Inc., 2006a. Fluent user’s guide.
Guo, H., Heyun, Liu, Xiaosong, Gu, 2010. Numerical simulation of the local collection efficiency on icing conductor. In: International Conference on Electrical and Control Engineering, pp. 2868–
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2871.
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Howe, J., 1991. Rotating Multicylinder Method for the Measurement of Cloud Liquid-Water Content and Droplet Size. U.S Army Cold Regions Research & Engineering Laboratory CRREL
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Report 91-2, Hanover, NH, 21 pp.
International Standardization Organization (ISO), 2000. International Standard for Atmospheric Icing of Structures, ISO 12494.
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Landau, L., Lifshitz, E.M., 1959. Fluid Mechanics. Pergamon Press, pp. 96-98.
ACCEPTED MANUSCRIPT Langmuir, I., Blodgett, K., 1946. A mathematical investigation of water droplet trajectories. Collected Works of I. Langmuir, vol. 10, Pergamon Press, 348–393.
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Makkonen, L, 1986. The effect of conductor diameter on ice load as determined by a numerical
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icing model. Proceedings, Third International Workshop on Atmospheric Icing of Structures
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(IWAIS), 81-89.
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liquid water content. J. Atmos. Oceanic Technol. 9, 258-263.
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Makkonen, L., Oleskiw, M.M., 1997. Small- scale experiments on rime icing. Cold Regions Sci. Technol. 25, 173 - 182.
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Makkonen, L., Stallabrass, J.R., 1987. Experiments on the cloud droplet collision efficiency of
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cylinders. J. Clim. Appl. Meteorol. 26, 1406-1411.
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by the Finite-Element Method. J. Atmos. Sci. 38, 1027-1036.
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ACCEPTED MANUSCRIPT Temkin, S., Mehta, H.K., 1982. Droplet drag in an accelerating and decelerating flow. J. Fluid. Mech. 116, 297-313.
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icing collision efficiency. Cold Regions Sci. Technol. 143, 52-58.
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Highlights
Fluid dynamics simulations on the droplet collision efficiency on cylinders were made at small collision efficiencies.
Icing wind tunnel experiments were made at small collision efficiencies to compare with the
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model results. It is confirmed that the history term in the droplet trajectory equations becomes relevant at
The model and experiments agree when the history term is included and the integral over
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small collision efficiencies.
the droplet size distribution used.
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Using this method, it becomes possible to estimate large cumulative rime ice accretions
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relevant in structural design.
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