Aircraft icing model considering both rime ice property

Aircraft icing model considering both rime ice property variability and runback water effect Xuan Zhang a, Xiaomin Wu a, *, Jingchun Min b, ** a. Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China. b. Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China. * Corresponding author at: Department of Thermal Engineering, Tsinghua University, Beijing 100084, China. Tel: +86-10-62770558, Email: [email protected] (X.M. Wu) ** Corresponding author at: Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China. Tel: +86-10-62783026, Email: [email protected] (J.C. Min)

Abstract: An improved one-dimensional model has been developed to describe the aircraft icing process, which can be divided into the dry and wet mode icing stages. Rime ice forms on aircraft skin at the dry mode icing stage while glaze ice grows on the rime ice and water film develops on the glaze ice at the wet mode icing stage. The model differs from the traditional icing models in its assumption that the rime ice is a kind of porous medium and its physical properties are initially affected by airflow parameters and then vary linearly with the rime ice thickness. Further, it differs from our previous icing model in its inclusion of runback water effect. Calculations are performed to analyze the ice accretion characteristics, and the results are presented and discussed in comparison with those given by the traditional model and our previous model. The results show that the rime ice property variability and runback water influence the heat conductions in the ice layer and water film and consequently the ice accretion characteristics. The model proposed in this research provides an alternative approach for modeling the ice accretion process. Key words: Aircraft icing model; Property-variable rime ice; Runback water; Ice accretion

Nomenclature PVRI PCRI RW B

c D

property-variable rime ice property-constant rime ice runback water equation coefficient phase speed of surface waves, m s1 airfoil leading edge diameter, m

f

vapor pressure constant (=27.03), Pa K -1 volume fraction

fF

freezing fraction

fi

interfacial friction factor

FT

interfacial shear stress acting on the water film at the air/water interface, N m-2

G

mass flow rate per unit of film width, kg m1 s1

hcon

convective heat transfer coefficient, hcon   e0  V  cw , W m2 K 1

he

equivalent heat transfer coefficient, W m2 K 1

k

thermal conductivity, W m1 K 1

e0

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Le

Lewis number

Lf

latent heat of fusion, J kg 1

Leva

latent heat of evaporation, J kg 1

MVD Pr

mean volumetric droplet diameter, μm pressure, Pa Prandtl number

qa

aerodynamic heat,

qcon

convection heat, hcon  taw  ta  , W m2

qeva

evaporative heat,  e0  taw  ta  , W m2

qd

sensible heat of incoming droplets, V  cw  taw  ta  , W m2

qk

kinetic energy of incoming droplets, V V 2  2 , W m2

qsource

sum of qa and qk , qa  qk , W m2

r

recovery factor Reynolds number temperature, °C

p

Re t v

 rh

V 2   2ca  , W m2

con

velocity, m s1

v

mean velocity, m s1

V

airflow velocity, m s1

Vw zc

water velocity at the air/water interface, m s1 thickness coordinate, m critical ice thickness given by the PVRI-RW model, m

Zc

critical ice thickness given by the PCRI-RW model, m

Greek Symbols  

collection efficiency

z

  

evaporation coefficient,

 0.622hcon Leva 

liquid water content (LWC), kg m-3 viscosity, N s m-2 density, kg m-3



time, s

Subscript 0 a c e g i m p aw r w

initial air critical equivalent glaze ice ice melting rime ice/glaze ice interface air/water interface rime ice water

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 c p Le  , 2/3

a

a

m s1

1 Introduction Aircraft icing presents a serious hazard for flight, which occurs when supercooled droplets in clouds impinge and stick on aircraft under natural icing conditions. It increases drag and reduces lift and controllability, even leads to serious flight accidents [1]. Developments of theoretical models for aircraft icing can help improve the accuracy of ice accretion prediction which eventually increases the flight safety. Extensive theoretical, numerical and experimental studies on aircraft icing have been conducted. Most theoretical and numerical models for aircraft icing start from the classical Messinger icing model [2]. In 1990s, NASA carried out plenty of experiments on aircraft icing [3], which provide good validations for simulations [4, 5] and numerical codes such as LEWICE, DRA, ONERA and FENSAP-ICE. Many researchers [6-8] modified the Messinger model [2] that considers only the energy balance between the latent heat release and heat transfer at gas/liquid interface but ignores the heat conductions in the ice layer and water film. Al-Khalil et al. [9], Myers et al. [10] and Du et al. [8] took into account the existence of rime ice as well as the effect of runback water in their models but they simplified the rime ice as property-constant glaze ice and assumed that the rime ice had the same physical properties as the glaze ice at constant. Our previous research [11] suggests that rime ice is actually a kind of porous medium and its effective properties (such as the volume fraction, density, and thermal conductivity) are initially influenced by the airflow parameters and then vary with the rime ice thickness, which affects the ice accretion characteristics. Fig. 1(a) gives a picture taken during icing wind tunnel test, it illustrates the variation of gray value along the ice thickness, which reflects the rime ice property variation. Our previous work [11] provides a model that considers the rime ice property variability but ignores the runback water effect, as a result, the water film thickness generated by the model shows a monotonous increase with time. Theoretical analyses [12] and experimental results [13] all indicate that the water film driven by airflow on aircraft surface has an ultimate thickness in an order of 100 microns. The present research is based on the work in Ref. [11], it improves our previous model by taking into account the runback water effect. The present paper seeks to provide an improved aircraft icing model that considers both the rime ice property variability and the runback water effect and to analyze their effects on the ice accretion characteristics. The results generated by the new model are compared with those by the traditional icing model that considers the runback water effect but treats the rime ice properties as constant, as in Refs. [6-8], and those by our previous model that assumes the rime ice properties to vary with the rime ice thickness but ignores the runback water effect [11]. For convenience, these three kinds of models are referred to as the PVRI-RW, PCRI-RW, and PVRI models, respectively, where PVRI expresses the property-variable rime ice, PCRI the property-constant rime ice, and RW the runback water. The present model (PVRI-RW model) differs from the traditional model (PCRI-RW model) in its assumption that the rime ice properties vary with the rime ice thickness, and it differs from our previous model (PVRI model) in its inclusion of the runback water effect. As described in Ref. [11], the initial stage during which a sublayer of rime ice forms is called the dry mode icing stage and the later stage during which glaze ice/water film develop is called the wet mode icing stage.

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The time when the dry mode transits to the wet mode is defined as the critical time and the ice thickness at that time as the critical ice thickness.

2 Theoretical model Fig. 1 (b) shows the basic physical model that consists of the rime ice layer, glaze ice layer and water film. The density varies linearly along the rime ice thickness but remains constant in the glaze ice layer and water film, it continues at the rime ice/glaze ice interface but jumps at the glaze ice/water interface. Couette flow occurs in the water film, implying that water flows with a linear velocity distribution as shown in Fig. 1 (b). The runback water effect is considered by including this water flow in the water film. Following assumptions are used in modeling the icing process: (1) the physical properties of rime ice vary linearly with the ice thickness while those of glaze ice and water are constant, (2) Couette flow takes place in the water film, causing a runback water effect, and (3) phase transition occurs isothermally.

2.1 Physical properties of rime ice As described in our previous paper [11], the initial rime ice density at the skin surface, r , 0 , is obtained from the empirical correlation [14-16],

 MVD0.82V 0.59 (1000 )0.21  6043   r  1000exp 0.15   1  2.65   , S  0.23 S   (10D)0.48  ta  

(1)

while the initial volume fraction, f r , 0 , is given by fr , 0 

 r , 0  a g  a

(2)

Since the rime ice is a kind of porous medium, its initial effective thermal conductivity, kr , 0 , can be calculated from kr , 0  kg 

2kg  ka  2 1  f r , 0   kg  ka 

(3)

2kg  ka  1  f r , 0   kg  ka 

which applies to a porous material [17, 18] and 1  f r , 0 is equal to the volumetric fraction of entrapped air. Assuming that the rime ice properties vary linearly with the rime ice thickness, the volume fraction, density, and effective thermal conductivity of rime ice can be represented by fr  B f z  fr , 0 , B f 

r  B z  r , 0 , B  k r  kg 

2kg  ka  2 1  f r   kg  ka  2kg  ka  1  f r   kg  ka 

1  fr , 0 zc

g   r , 0 zc

, 0  z  zc

(4)

, 0  z  zc

 Bk z  kr , 0 , Bk 

kg  kr , 0 zc

(5) , 0  z  zc

(6)

2.2 Dry mode icing and critical point Assuming that the heat conduction is at a quasi-stable state at the dry mode icing stage, the temperature profile in rime ice is

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 B  ln  k z  1 kr , 0  t tr   tp  ts   s  Bk  ln  zc  1  kr , 0 

(7)

The time variations of the rime ice thickness and rime ice growth rate can be obtained from the mass balance at the dry mode icing stage zr 

2V  B  r2, 0  r , 0 B

dzr V   d B z  r , 0

(8) (9)

The critical time and critical ice thickness are the same as those in Ref. [11]

c

B z 

 r , 0   r2, 0 2

 c

zc  Z c

2V  B

kg  k r , 0

kg ln  kg / kr , 0 

(10) (11)

where Z c is the critical ice thickness generated by the PCRI-RW model, given by kg  tm  ts 

Zc =

V  Lf +qsource  he  tm  ta  The detailed derivations for the above formulas can be found in Ref. [11].

(12)

2.3 Wet mode icing Assuming that the heat conduction is at a quasi-stable state at the wet mode icing stage, the temperature profile in glaze ice is

tg 

tm  tp zg

 z  zc   tp

(13)

2.3.1 Governing equations In the water film, the flow driven by the air fluid is simplified as one-dimensional Couette flow. The one-dimensional momentum and energy equations therefore are d 2 vw 0 (14) dz 2 2 d 2tw  dvw  kw 2   w  Energy equation (15)  dz  dz  The ice growth rate at the wet mode icing stage can be related with the airflow parameters as below

w

Momentum equation

g

where f F

dzg

 f FV  (16) d is the freezing fraction which is the ratio of frozen-to-collected water mass per unit time, it is

equal to unity at the dry mode icing stage. 2.3.2 Boundary conditions At the rime ice/glaze ice interface, the temperature, heat flux, and physical properties of the rime ice are equal to those of the glaze ice, i.e. 5 / 13

tr  z  zc   tg  z  zc  , kr

r

z  zc

 g , kr

tr z z  zc

 kg z  zc

tg z

(17) z  zc

 kg

(18)

where zc is the critical ice thickness. At the glaze ice/water interface, the energy equation is tg

tw d z z The water film temperature and flow boundary conditions are

g Lf

dzg

 kg

 kw

tw  z  zi   tm , vw  z  zi   0

tw  z  zi  zw   tf , vw  z  zi  zw   Vw

(19)

(20) (21)

where zi is the total ice thickness, i.e. zi  zr  zg , and tf and Vw are the water temperature and velocity at the air/water interface, respectively. At the air/water interface, the heat flux satisfies kw

tw z

  qcon  qeva  qd    qa  qk 

(22)

z  zi  zw

which includes: (1) convective heat transfer, qcon  hcon  taw  ta  , where hcon is the convective heat transfer coefficient for a cylinder whose diameter is equal to the airfoil leading edge diameter [19]; (2) evaporative heat loss, qeva   e0  taw  ta  [20]; (3) incoming droplet cooling, qd  V  cw  taw  ta  ; (4) aerodynamic heating, qa   rhconV 2   2ca  , in which r is the recovery factor, given by r  Pr n , where n  1 / 2 for laminar flow and n  1 / 3 for turbulent flow [21]; (5) incoming droplet kinetic energy,

qk  V V 2  2 . So, the unified equations are qcon  qeva  qd  he tw  ta  and qa  qk  qsource . 2.3.3 Water film The solution of Eq. (14) with the boundary conditions, Eqs. (20) and (21), defines a linear velocity profile

vw 

FT

w

 z  zi 

; Vw 

FT zw

w

(23)

where FT is the interfacial shear stress acting on the water film at the air/water interface, given by [22] 1 2 (24)  a Va  c  , fi  0.0055  2.6  105 Rew 2 in which f i is the interfacial friction factor, c is the phase velocity of surface waves, and Rew is the FT  fi

Reynolds number of the water film defined by Rew  w vw zw w . As in Cheremisinoff et al. [23], it is reasonable to assume Va

c . Eq. (24) can thus be simplified as

1 (25)  aVa2 , fi  0.0055  2.6  105 Rew 2 In the present model, the Reynolds number at the stagnation point can also be expressed as  v z G 1  f F  V  (26) Rew  w w w   FT  fi

w w w where G is the mass flow rate per unit of film width in the direction perpendicular to the cross-section of 6 / 13

ice layer shown in Fig. 1, and f F is the freezing fraction defined by

g dzg (27) V  d Integrating the flow profile, Eq (23), over the film thickness, one can get the mean water film velocity fF 

FT zw (28) w 2w Combining Eq. (26) with Eq. (28) yields a relation between the water film thickness and time, vw  

z  zi  zw

z  zi

vw dz  

z  zi  zw

FT

z  zi

 z  zi  dz 

g dzg  2w 1  f F  V  2w G 2wV     1   w FT w FT w FT  V  d  For the water film, Eq. (15) can also be expressed using Eq. (23) as zw 

F  d 2t kw 2w   w  T  dz  w 

(29)

2

(30)

and the solution of Eq. (30) with the boundary conditions, Eqs. (20) and (21), determines the water film temperature distribution t w  tm 

taw tm F2  z  zi   T  zi  zw  z   z  zi  zw 2k w  w

(31)

2.3.4 Glaze ice The relation between the glaze ice thickness and time can be obtained by combining the glaze ice/water interface governing equation, Eq. (19), with the temperature profiles, Eqs. (13) and (31)

g Lf

dzg d

 kg

tg z

 kw

tm  tp tw t t z F2  kg  kw aw m  w T z zg zw 2w

(32)

while the relation between the water film thickness and time is given by Eq. (29). Note that Eq. (32) is a nonlinear differential equation, and the temperatures at the rime ice/glaze ice and air/water interfaces, tp and taw , are unknown. Combining the boundary condition, Eq. (17), with the temperature profiles, Eqs. (7) and (13), yields Zc tm  ts zg  zc zc tp  zc  ts zg  Z c zg  Z c

(33)

while combining Eqs. (22) and (31) gives the water film temperature at the air/water interface

 heta  qsource  zw  kw tm  taw 

kw  he zw

zw 2 FT 2 2w

(34)

2.4 Comparison with the PCRI-RW model [7, 8] As stated in Ref. [11], when the initial rime ice density, r , 0 , approaches the glaze ice density, g , the temperature distribution in the rime ice and the critical ice thickness given by the PVRI-RW model approaches those by the PCRI-RW model, i.e.

lim tr   tp  ts 

kr , 0  kg

z  ts zc

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(35)

lim zc 

kr , 0 kg

kg  tm  ts 

V  Lf +qsource  he  tm  ta 

 Zc

(36)

So, the PVRI-RW model will reduce to the PCRI-RW model if the initial rime ice density is taken to be the same as the glaze ice density.

2.5 Comparison with the PVRI model [11] The equation of energy balance at the glaze ice/water interface in the PVRI model [11] is dz t t q  h t  t  g Lf g  kg m p  source e m a he d zg zw  1 kw while that in the present PVRI-RW model is Eq. (32), which can be rewritten as

(37)

  z F 2 qsource  he tm  ta   1  w T g Lf  kg   1  (38) he he d zg   2w zw  1 z  1 w   kw kw   It is useful to note that the water film thickness in Eq. (38) is different from that in Eq. (37), the existence dzg

tm  tp

of water runback greatly reduces the water film thickness, causing a reduced water film thermal resistance and consequently an increased ice growth rate, in addition, the water flow in the water film can promote the heat transfer in it, which also contributes to the increase of the ice growth rate. As a result, the PVRI-RW model yields a larger ice growth rate than the PVRI model.

3 Results and discussion Eqs. (37) and (38) are both nonlinear differential equations governing the glaze ice thickness change with time. After the critical time (  c ) and critical ice thickness ( zc ), the Runge-Kutta method is used to numerically solve these nonlinear differential equations, with the time step being set as 0.05 s. There is a good convergence with the numerical scheme.

3.1 Model validation As stated in our previous paper [11], the gray value profile of the rime ice shown in Fig. 1(a) [24] provides a basis for the assumption that the rime ice volume fraction varies linearly with the rime ice thickness. Myers and Hammond [6, 7] experimentally investigated the ice accretion on NACA0012 airfoil having a dimension of 40 cm  4.8 cm with airflow parameters of MVD  20 μm , V  90 m/s ,

  1 g/m3 and ta  10°C , and they reported the measured rime ice thickness to be 2~3 mm. Using these parameters, our model yields a rime ice thickness of 2.50 mm, which is consistent with the measurement. Anderson and Feo [13] conducted tests on water film thickness on a conductance sensor at ambient temperatures well above freezing, and they reported a result of 55~150 μm. Calculations are made to predict the water film thickness evolutions using various models and the obtained results are presented in Fig. 2 (a), which shows that as time goes by, the water film thickness generated by the PVRI-RW model approaches a constant of about 100 μm, which falls in the range of 55~150 μm. The PCRI-RW model gives a similar result, implying that the rime ice has only a minimal influence on the water film thickness, which is affected mainly by the airflow. The above comparisons support the reliability of our model.

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3.2 Basic ice accretion characteristics Calculations are made for MVD=20 μm , V  40 m/s ,   1 g/m3 ,   0.55 , ta  ts  12°C and

tm  0°C , these parameters are taken based on the data in Refs. [3, 5]. The results generated by the PCRI-RW, PVRI-RW, and PVRI models are presented in Fig. 2, which compare the evolutive ice layer and water film thicknesses, ice growth rates, water freezing fractions, and the temperature profiles in the ice layers and water films at   600 s . In Fig. 2 (b), the asterisks express the critical point. The critical time and critical ice thickness given by the PVRI model are the same as those by the PVRI-RW model because they share the same dry mode icing stage. However, the PVRI-RW model yields a shorter critical time and a smaller critical ice thickness than the PCRI-RW model because the PVRI-RW model treats the rime ice as a porous medium that has a smaller effective thermal conductivity and density than the glaze ice while the PCRI-RW model treats the rime ice as the glaze ice and assumes that the rime ice has the same properties as the glaze ice at constant. From Fig. 2 (c) and (d), the ice thickness and ice growth rate given by the PVRI-RW model is larger than those by the PCRI-RW model at the early icing stage but they turn to be smaller at the later stage. The reason is that: for    c , the density of rime ice forming at the dry mode icing stage is smaller than that of glaze ice, so the PVRI-RW model yields a larger ice growth rate and a thicker ice layer; when    c , the rime ice density at the rime ice/glaze ice interface is equal to the glaze ice density, and so is the ice growth rate; after that time, the thicker ice and water film given by the PVRI-RW model causes a larger thermal resistance and consequently a lower ice growth rate, with time going by, the ice thickness given by the PCRI-RW model exceeds that by the PVRI-RW model. From Fig. 2 (e), the water freezing fractions generated by these two models show variations similar to those of the ice growth rates. As time passes, the effect of rime ice layer on the total thermal resistance becomes weaker, the ice growth rate given by the PVRI-RW model thus approaches that by the PCRI-RW model, this also applies to the water freezing fraction. The ice thickness, ice growth rate, and water freezing fraction given by the PVRI-RW model are the same as those by the PVRI model at the dry mode icing stage but they become larger than those by the PVRI model at the wet mode icing stage as shown in Fig. 2. There are two reasons for the latter: (1) according to Eqs. (37) and (38) and the relative analysis, the water flow promotes the heat transfer in the water film which works to accelerate the latent heat release and consequently increase the ice growth rate, and (2) the PVRI model yields a thicker water film due to its ignorance of the runback water effect and consequently a greater thermal resistance that acts to reduce the ice growth rate. As time passes, the water film thickness given by the PVRI-RW model initially increases and then approaches a constant while that by the PVRI model monotonously increases, causing an increased thermal resistance, as a result, the differences of the ice thickness, ice growth rate, and water freezing fraction between these two models broaden. From Fig. 2 (e), the rime ice temperature profiles generated by the PVRI and PVRI-RW models are logarithmic curves due to the variations of the rime ice physical properties with the ice thickness while that by the PCRI-RW model is a straight line. The water film temperature profiles given by these three models 9 / 13

differ from one another due to different water film thicknesses. The water film temperature profiles given by the PVRI-RW and PCRI-RW models both are parabolic, this can be realized from Eq. (31). We note that Ref. [11] provides a more detailed discussion on the basic ice accretion characteristics, which can serve as a useful supplement to the above analyses.

4 Conclusions With inclusion of the runback water effect and assumption that the rime ice physical properties vary linearly with the rime ice thickness, an improved aircraft icing model named the PVRI-RW model has been developed, and the effects of the airflow parameters on the ice accretion characteristics are analyzed. The results are compared with those by the traditional PCRI-RW model and our previous PVRI model. The following conclusions can be drawn: (1) The rime ice physical properties and runback water effect are affected by the airflow parameters, they all influence the heat conductions in the ice layer and water film as well as the ice accretion process. (2) The differences between the PVRI-RW and PCRI-RW models depend on the rime ice properties. The ice thickness and ice growth rate given by the PVRI-RW model is larger than those by the PCRI-RW model at the early icing stage but they turn to be smaller at the later stage. (3) The PVRI-RW and PVRI models share the same dry mode icing stage, with the former yielding a larger ice growth rate and a thicker ice layer than the latter at the wet mode icing stage. The runback water needs to be taken into account due to the obvious differences between these two models. (4) The PVRI-RW model provides an alternative approach for modeling the ice accretion process, it can best describe the heat conductions in the ice layer and water film, the water flow in the water film, and the ice accretion characteristics.

5 Acknowledgements This research is funded by National Key Basic Research Program of China (No. 2015CB755800).

6 References [1] Y.H. Cao, Z.L. Wu, Y. Su, Z.D. Xu, Aircraft flight characteristics in icing conditions, Prog. Aerosp. Sci. 74 (2015) 62-80. [2] B.L. Messinger, Equilibrium temperature of an unheated icing surface as a function of air speed, J. Aeronaut. Sci. 20 (1) (1953) 29-42. [3] J. Shin, T.H. Bond, Results of an icing test on a NACA 0012 airfoil in the NASA Lewis Icing Research Tunnel, NASA TM-105374, Lewis Research Center, Cleveland, Ohio, 1992. [4] R.J. Kind, M.G. Potapczuk, A. Feo, C. Golia, A.D. Shah, Experimental and computational simulation of in-flight icing phenomena, Prog. Aerosp. Sci. 34 (5-6) (1998) 257-345. [5] X. Yi, Numerical computation of aircraft icing and study on icing test scaling law (PhD thesis), China Aerodynamics Research and Development Center, Mianyang, Sichuan, 2007. [6] T.G. Myers, Extension to the Messinger model for aircraft icing, AIAA J. 39 (2) (2001) 211-218. [7] T.G. Myers, D.W. Hammond, Ice and water film growth from incoming supercooled droplets, Int. J. Heat Mass Transfer 42 (12) (1999) 2233-2242. [8] Y.X. Du, Y.W. Gui, C.H. Xiao, X. Yi, Investigation on heat transfer characteristics of aircraft icing including runback water, Int. J. Heat Mass Transfer 53 (19–20) (2010) 3702-3707. [9] K.M. Al-Khalil, T.G. Keith, K.J. De Witt, Development of an improved model for runback water on aircraft surfaces, J. Aircr. 31 (2) (1994) 271-278. [10] T.G. Myers, C.P. Thompson, Modeling the flow of water on aircraft in icing conditions, AIAA J. 36 (6) (1998) 1010-1013. [11] X. Zhang, J.C Min, X.M Wu, Model for aircraft icing with consideration of property-variable rime ice, Int. J. Heat Mass Transfer 97 (2016) 185-190. [12] A. Rothmayer, J.C. Tsao, Water film runback on an airfoil surface, in: 38th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 2000. [13] D.N. Anderson, A. Feo, Ice-accretion scaling using water-film thickness parameters, NASA CR-2003-211826, Ohio Aerospace Institute, Brook Park, Ohio, 2003.

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[14] W.B. Wright, Users manual for the improved NASA Lewis ice accretion code LEWICE 1.6, NASA CR-198355, HYMA, Inc., Brook Park, Ohio, 1995. [15] M. Rios, Icing simulations using Jones' density formula for accreted ice, in: 29th Aerospace Sciences Meeting, Reno, Nevada, 1991. [16] K.F. Jones, The density of natural ice accretions, in: Fourth International Conference on Atmospheric Icing of Structure, EDF, Paris, 1988, pp. 114–118. [17] H.T. Aichlmayr, F.A. Kulacki, The effective thermal conductivity of saturated porous media, in: G.A. Greene, J.P. Hartnett Avram (Eds), Advances in Heat Transfer, Elsevier, 2006, pp. 377-460. [18] P. Schwerdtfeger, The thermal properties of sea ice, Journal of Glaciology 4 (36) (1963) 789-807. [19] F.P. Incropera, A.S. Lavine, D.P. DeWitt, Fundamentals of Heat and Mass Transfer, seventh ed., John Wiley & Sons, New York, 2011. [20] P.R. Lowe, An approximating polynomial for the computation of saturation vapor pressure, J. Appl. Meteorol. 16 (1) (1977) 100-103. [21] J.H. Lienhard, A Heat Transfer Textbook, fourth ed., Phlogiston Press, Cambridge, 2012. [22] G.V. Tsiklauri, P.V. Besfamiliny, Y.V. Baryshev, Experimental study of hydrodynamic processes for wavy water film in a cocurrent air flow, in: F. Durst, G.V. Tsiklauri, N.H. Afgan (Eds), Two-Phase Momentum, Heat and Mass Transfer in Chemical, Process, and Energy Engineering Systems, Hemipshere Publishing Corp., Washington, DC, 1979, pp. 357-372. [23] N.P. Cheremisinoff, E.J. Davis, Stratified turbulent-turbulent gas-liquid flow, AIChE J. 25 (1) (1979) 48-56. [24] Y.X. Du, Y.W. Gui, C.H. Xiao, X. Yi, Investigation of heat transfer characteristics of aircraft icing under runback water, J. Aerosp. Power 24 (09) (2009) 1966-1971.

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100

A

Glaze Ice

Pixel

75

50

25

Rime Ice 0 100

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Horizontally Averaged Gray Value

Cylinder

(a)

z

ta

Air

taw

zw

vw

Water

tm Glaze ice

zg

tp

zc zr (b)

O ts

Skin

t

 r, 0  r g  w

Rime ice

Fig. 1 Icing model: (a) Variation of gray value with ice thickness [11], (b) Physical model including property-variable rime ice and runback water.

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

1.0

16

 = 1 g/m3, V = 40 m/s, , ta = ts = -12C

0.8

PCRI-RW PVRI-RW PVRI

 = 1 g/m3, V = 40 m/s, , ta = ts = -12C PCRI-RW PVRI-RW PVRI

12

zw /mm

zi /mm

0.6

8

0.4 (227.39, 5.46)

4

0.2 (75.89, 2.82)

0.0

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400

600

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-2

 = 1 g/m3, V = 40 m/s,  , ta = ts = -12C

5 4

 = 1 g/m3, V = 40 m/s, , ta = ts = -12C

1.0

PCRI-RW PVRI-RW PVRI

PCRI-RW PVRI-RW

0.8

3

PVRI

fF

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c = 75.89 s

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= 1 g/m3, V = 40 m/s, , ta = ts = -12C 0  = 600 s PCRI-RW PVRI-RW PVRI

-2 -4 t /C

(dzi/d) /(mm/s)

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-6 -8

-10 -12

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zc =2.82 mm

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4

8 z /mm

12

(e) Fig. 2 Ice accretion characteristics with comparisons among various models: (e) Water film thicknesses, (b) Ice thicknesses, (c) Ice growth rates, (d) Water freezing fractions, (e) Temperature profiles at   600 s .

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800