Numerical Simulation of Unsteady Conjugate Heat Transfer...

Research ArticleNumerical Simulation of Unsteady Conjugate Heat Transfer ofElectrothermal Deicing Process

Zuodong Mu , Guiping Lin, Xiaobin Shen , Xueqin Bu, and Ying Zhou

Laboratory of Fundamental Science on Ergonomics and Environmental Control, Beihang University, Beijing 100191, China

Correspondence should be addressed to Xiaobin Shen; [email protected]

Received 2 June 2017; Accepted 28 November 2017; Published 29 January 2018

Academic Editor: Paul Williams

Copyright © 2018 Zuodong Mu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A novel 3-D unsteady model of in-flight electrothermal deicing process is presented in this paper to simulate the conjugate massand heat transfer phenomena of water film runback, phase change, and solid heat conduction. Mathematical models of waterfilm runback and phase change are established and solved by means of a loosely coupled method. At the current time step, solidheat conduction, water film runback, and phase change are iteratively solved until the heat boundary condition reachesconvergence, then the temperature distribution and ice shape at the moment are obtained, and the calculation of the next timestep begins subsequently. A deicing process is numerically simulated using the present model following an icing tunnelexperiment, and the results match well with those in the literatures, which validate the present model. Then, an in-flight deicingprocess is numerically studied to analyze the effect of heating sequence.

1. Introduction

The water droplet in clouds may remain in a liquid state evenif the temperature is below freezing point due to the surfacetension and a lack of condensation nucleus. Supercooledwater droplet would solidify when impinging on windwardsurfaces of aircraft, which would cause the deterioration inthe aerodynamic performance due to the change of aerody-namic configuration [1]. Aircraft icing would impose seriousadverse effects on flight safety, which has long been recog-nized as an important issue to prevent.

In view of the serious threats that ice accretion wouldimpose on flight safety, ice protection methods must beapplied to prevent or control the ice accretion. Hot air anti-icing method is widely applied in commercial jets. The bleedair from engine compressor impinges on the structure to heatthe surface, so that the water evaporates or stays in a liquidstate rather than freezes. Electrothermal ice protectionmethod uses heating pads, which are incorporated in themultilayer structure, to heat the surfaces. Electrothermalpads are available in both anti-icing mode with a constantpower density and deicing mode with a periodic powerdensity. Besides, some other deicing methods have been

developed, such as the thermomechanical expulsion deicingmethod [2], while they are not widely applied. A largeamount of bleed air is needed in hot air anti-icing system,which would significantly affect the performance of anengine, especially during the takeoff or landing period.Meanwhile, the jet flow would cause excessive temperatureat the impingement point, which may damage the compositematerials that are widely used in a modern aircraft [3] or eventhe aluminum skin. The electrothermal ice protectionmethod has a higher heating efficiency and a lower powerdensity, which results in the advantages in structure safetyand energy efficiency. As the concepts of more-electric-aircraft and all-electric-aircraft are increasingly employed ina modern aircraft design, the electrothermal ice protectionmethod is drawing more attention. The ice protection systemof B-787 [4], which is based on the electrothermal method,serves as an example.

Since the icing tunnel experiments and the flight experi-ments are complex, expensive, and not able to cover all envi-ronment conditions, the investigation of a deicing process byan experimental method is quite limited and seldom reportedto date. The numerical simulation method, which is time-efficient and cost-effective, becomes an important tool for

HindawiInternational Journal of Aerospace EngineeringVolume 2018, Article ID 5362541, 16 pageshttps://doi.org/10.1155/2018/5362541

http://orcid.org/0000-0001-9684-2250http://orcid.org/0000-0002-0735-5774https://doi.org/10.1155/2018/5362541

the design of the ice protection system. The in-flightdeicing process is a conjugate mass and heat transfer phe-nomenon, which is very complex and consists of a varietyof coupled processes such as the air-droplet flow, waterfilm runback and phase transition, and multilayer solidheat conduction. The phase state varies with both spatiallocations and time. Due to the periodic power density, spatialdistribution of heaters, and sustained droplet impingement,phase transition phenomena, such as evaporation, solidifica-tion, liquefaction, and sublimation, may coexist on thesurface of the protection area, which makes it complex andtypically unsteady.

The early study of electrothermal deicing concentratedon the solution of multilayer structure heat conduction.Stallabrass [5] numerically analyzed the heat conductionduring the deicing process, focusing on the effects of multi-layer materials on the distribution of skin temperature,especially the effects of inner and outer insulating layers.Then, the latent heat was introduced to simulate the phasechange during the deicing process by Baliga [6]. As theresearch went in depth, Marano [7] and Roelke et al. [8]developed the “enthalpymethod,” in which the unified energyequation was applied in the whole computational domain andthe phase interface was determined by enthalpy distribution.Based on the enthalpy method, plenty of researches wereconducted to simulate the phase transition process. Chao[9], Leffel [10], and Masiulaniec [11] applied the method to2-D cases and analyzed the effects of surface curvature andheating pad spatial distribution. Chang [12, 13] numericallystudied the 2-D deicing process by a finite volume methodand analyzed the effects of the heating sequence and powerdensity. Yaslik et al. [14] studied a 3-D deicing process, usinga Douglas method to discretize the heat conduction equation.Xiao [15] applied a porous medium method to simulate theice melting process. As is briefly reviewed above, researchershave done much work on the study of multilayer structureheat conduction and ice melting process based on enthalpymethod, while the study of the in-flight deicing process, whichinvolves sustained droplet impingement and water filmrunback, is quite rare.

Wright et al. [16] developed a 2-D deicing model, whichwas based on the classic Messinger mass and energy balancemodels [17]. The mass and energy balance equations tookinto account the input water caused by droplet impingement,

but the water film runback mechanism was not taken intoconsideration. Habashi et al. [18–20] developed a conjugatemodel for the in-flight deicing process. The airflow anddroplet impingement were calculated as initial solutions,and the solid conduction was solved, which was coupled witha water phase transition in a loosely coupled way. Finite ele-ment solvers were applied to the solution as the modules ofthe commercial software FENSAP-ICE, while the detailedsolution method was not available in the open literature.

As electrothermal deicing method is increasinglyemployed in the new generation of aircraft, the study on themechanism and simulation methods is urgently needed.Whilemost current studies on the deicing process concentrated onthe multilayer structure heat conduction and ice melting pro-cess, studies available on the in-flight deicing were quite rare.

This paper focuses on the conjugate heat transfer mecha-nism of the in-flight electrothermal deicing. A novel 3-Dunsteady model is established based on the water film run-back dynamic mechanism and phase transition thermody-namic model and solved by a loosely coupled method.Using the present model, the deicing process is numericallyinvestigated, which contributes to a better understanding ofelectrothermal deicing so as to guide the design and optimi-zation of an aircraft ice protection system.

2. Mathematical Model

The in-flight deicing process is a conjugate mass and heattransfer process, which involves air-droplet flow, solid heatconduction, water film runback, and the phase transition.The mass and energy balance, as is briefly shown inFigure 1, is determined by a variety of factors including con-vection, droplet impingement, evaporation, solidification,heat conduction, and film runback. Due to the periodicpower density and spatial distribution of heaters, the phasetransition process is typically unsteady.

2.1. Air-Droplet Flow. Since the volume fraction of droplet isvery small, typically under 10−6 for icing conditions, the air-droplet flow can be solved by a one-way coupled method[21]. Reynolds-averaged Navier–Stokes equations (RANS)are applied to solve the air flow field. The mass and momen-tum equations are expressed as (1), and the energy equationis expressed as (2):

∂ρa∂t

+ ∂ ρauixi

= 0,1

∂∂t

ρaui +∂∂xj

ρauiuj = −∂p∂xi

+ ∂∂xj

μ∂ui∂xj

+∂uj∂xi

−23 δij

∂ul∂xl

+ ∂∂xj

−ρaui′uj′ ,

∂∂t

ρaE +∂∂xi

ui ρaE + p =∂∂xj

λ +cpμ

Pr∂T∂xj

+ ∂∂xj

ui μ + μt∂ui∂xj

+∂uj∂xi

−23 δi,j

∂ul∂xl

, 2

2 International Journal of Aerospace Engineering

where the Reynolds stress is defined by Boussinesq as

−ρui′uj′ = μt∂ui∂xj

+∂uj∂xi

−23 ρk + μt

∂uk∂xk

δij, 3

where μt is the turbulent viscosity, ui is the average velocity,and k is the turbulence kinetic energy.

The continuity and momentum equations of droplet floware expressed as [22]

∂ ρα∂t

+ ∇ ⋅ ραu = 0, 4

∂ ραu∂t

+ ∇ ⋅ ραuu = ραK ua − u + ραF, 5

where ρ is the density of water, α is the droplet volumefraction, u is the velocity vector of droplet, ua is the velocityvector of air, ραF is the external body forces exerted ondroplet, such as the gravity or inertial force, and K is theair-droplet momentum exchanger coefficient defined as

K =18μf dragρd2p

, 6

where μ is the dynamic viscosity of air, dp is the diameter ofdroplet, and f drag is the drag function. The Schiller andNaumann model is adopted here.

f drag =CDRer24

CD =24 1 + 0 15Re0 687r

Rer, Rer ≤ 1000,

0 44, Rer > 1000

7

Rer is the relative Reynolds number and is given as

Rer =ρa∣ua − u∣dp

μ8

2.2. Water Film Runback and Phase Transition. The differen-tial form of continuity equation for runback water film on thesurface of ice protection area is expressed as

∂ρ∂t

+ div ρv =mimp −mice −mevap, 9

where v is the velocity of water film,mimp is the water mass ofdroplet impingement, mice is the icing rate, and mevap is theevaporation rate. The water film is incompressible, and thedevelopment of water film is quick due to a quite small thick-ness. Therefore, the unsteady water mass term is neglected,and the continuity equation is derived by integrating thedifferential form in a control volume.

〠min,n +mimp =〠mout,n +mice +mevap, 10

where ∑min,n is the total mass of water entering the currentcontrol volume from adjacent ones per unit time, ∑mout,n isthe total mass of water flowing out of the control volume,and the mass flux through a face n can be expressed as

mout,n = ρlnh v ⋅ nn , 11

where h is the film thickness, ln is the edge length, and nnrepresents the unit vector normal to face n.

The mass flux of droplet impingement is determined bylocal collection efficiency β, as expressed as

mimp = u∞ ⋅ LWC ⋅ β ⋅ A, 12

where u∞ is the velocity of droplet at far field, LWC is theliquid water content, and A is the area of control volume.

The evaporation rate is determined by the Chilton-Colburn analogy theory where the convective mass transfer

Heat conduction,water film runback, and phase transition

Solidificationand melting

Heat generationand conduction

Heater

Convection

EvaporationDroplet

impingement

Film runback

Air flow

Droplet

Figure 1: Mass and energy balance of deicing process.

3International Journal of Aerospace Engineering

coefficient kc is obtained from the convective heat transfercoefficient hc as shown below.

hccp

Pr2/3 = kcSc2/3, 13

where Sc is the Schmidt number. The convective heat trans-fer coefficient hc is calculated according to the surface heatflux and the temperature difference.

hc =q

A ⋅ T − T∞ 1 + f rec γ − 1/2 Ma2, 14

where f rec is the recovery coefficient; γ is the air specific heatratio, of which the value is 1.4; and Ma is the Mach numberof air flow. Then the evaporation rate is derived as

mevap =hcAcp,air

⋅MWwaterMWair

⋅PrSc

2/3

⋅pv,satpT

⋅TTT

⋅pTp

1/γ−pv,∞,satrh

p∞,

15

whereMW is the air molecular weight, cp,air is the air specificheat, pv,sat is the vapor saturated pressure, pT is the totalpressure, TT is the total temperature, and rh is the relativehumidity, of which the value on water film surface is 1.

The momentum equation of water film runback is givenby incompressible Navier–Stokes equation as

∂∂t

ρv + ∇ ⋅ ρvv + pI = ∂∂y

μ∂v∂y

+ ρg, 16

where p is the pressure, I is the unit tensor, y is the coordinatenormal to surface, and g is the gravitational acceleration.

The air-film boundary condition is defined as

μ∂v∂y

= τ + σκ, y = h, 17

where τ is the shear stress, σ is the coefficient of tension, andκ is the water film surface curvature which is given by

κ = − ∂2h

∂x2j1 + ∂h

∂xj

2 −3/2

18

The solid-film boundary is depicted under the nonslipboundary condition.

v = 0,y = 0

19

Previous studies suggest that the effects of pressure gradi-ent should only be considered for water film with a largethickness [23]. During the deicing process, the water film isvery thin; therefore, the terms of gradient and gravity arenegligible, of which the effect is slight compared with otherterms such as shear stress. The tangential velocity gradientis also very small; therefore, the momentum diffusion termis negligible [24]. The value of surface curvature is generallyvery small for film which entirely wets the surface, and theshear stress is the dominant factor at the air-film interface.Under such condition, the momentum equation and bound-ary condition of water film are simplified as

∂∂y

μ∂v∂y

= 0,

μ∂v∂y

= τ, y = h,

v = 0, y = 0

20

The velocity distribution normal to the surface isderived as

v = yμτ 21

Air flow Droplet flow

Unsteady time step

Update ice accretion state

Coupled iteration

Icing/melting rateBoundary convergence

Water film runbackand phase transition

Multilayer heatconduction

Figure 2: Diagram of coupling solution method.


The average velocity V is obtained by integrating theabove equation along the film thickness direction, asexpressed as

V = h2μ τ 22

The energy balance of water film is described by the fol-lowing differential equation:

∂ ρcwT∂t

+ div ρcwTV =Himp + Eice −Qc −Hevap +Qcond23

Integrating the above equation, the energy balance in acontrol volume is expressed as

ρAhcw T − TpreΔt +〠n

Hout,n −〠n

Hin,n =H imp + Eice −Qc

−Hevap +Qcond,24

where cw is the specific heat of water, T and Tpre are the tem-perature at current time step and previous time step, Qcond isthe deicing heat flux from solid, obtained from the calcula-tion of solid heat conduction during the iterations, and

Start Calculate the airflowCalculate thedroplet flow Assume Ts = T0

Use Ts to calculate f, solveenergy balance equition

Find the stagnation point andboundary

Interpolationand store data

Export localcollectionefficiency

Export shearstress and heatflux

0

Hin,n is the energy carried by runback water entering thecurrent control volume through the face n, and it can beexpressed as

Hin,n =min,ncw T in,n − T0 25

Himp is the energy of impinging water, which consists oftwo parts: the internal energy and the kinetic energy, asexpressed below.

Himp =mimp cw T∞ − T0 +12 u

2∞ 26

Qc is the convective heat flux obtained by the con-vective heat transfer coefficient hc and the temperaturedifference between surface temperature T and recovertemperature Trec.

Qc = hcA T − T rec 27

The latent heat of solidification Eice, the out-flow energyHout,n, and the latent heat of evaporation Hevap are relatedto the phase condition of the control volume, which arediscussed as follows by considering the freezing fractionf , namely the ratio of icing mass to the total enteringwater mass.

If f = 0 and the surface is clean, which means currentlythat there is no ice accretion, the water in the control volumeremains at a liquid state, icing rate is zero, and the energy ofrunback water and evaporation can be expressed as

mice = 0,Eice = 0,

Hout,n =mout,ncw Ts − T0 +12mout,nV

2,

Hevap =mevap Levap + cw Ts − T0 ,

28

where Levap is the latent heat of water evaporation.If f = 0 and there is ice accretion in the current volume,

the ice melts at a liquid-solid mixed state and the controlvolume remains at the phase transition temperature.

Ts = T0,Eice =mmeltLs,

Hout,n = 1/2mout,nV2,Hevap =mevapLevap,

29

where mmelt is the melting rate and Ls is the solidificationlatent heat.

If 0< f< 1, part of water freezes, and the control volume isat the phase transition temperature.

Ts = T0,Eice =miceLs,

Hout,n = 1/2mout,nV2,Hevap =mevapLevap

30

If f = 1, all the water freezes and the energy terms can beexpressed as

Eice =mice Ls + cice T0 − Ts ,mout,n = 0,Hout,n = 0,Hevap =mevap Lsub + cice Ts − T0 − Ls ,

31

Heater

Heater

12

3

4

5

Erosion shield0.2 mm

Elastomer0.56 mm

Fiberglass0.89 mm

Silicone foam3.4 mm

6 7

Figure 4: Material and structure of ice protection area.

Table 2: Material properties of NASA experiment.

MaterialDensity(kg/m3)

Heat conductivity(W/mK)

Heat capacity(J/kgK)

Erosionshield

8025.25 16.26 502.4

Elastomer 1383.96 0.2561 1256.0

Fiberglass 1794 0.294 1570.1

Siliconefoam

648.75 0.121 1130.4


where cice is the specific heat of ice and Lsub is the sublimationlatent heat.

2.3. Heat Conduction. The unsteady heat conduction throughthe multilayer materials is expressed as

∂H∂t

= ∇ ⋅ λ∇T + S, 32

where H is the enthalpy of the material, λ is the thermal con-ductivity, and S is the heat source term. For electrothermaldeicing process, the source term is determined by the spatiallocation of heaters and the power sequence.

The Dirichlet heat boundary condition of heat conduc-tion is provided by the solution of water film runback andphase transition during the coupled solution. In return, thedeicing heat flux is calculated and sent to the calculation ofwater film. The deicing heat flux is obtained at the boundaryof a solid structure as shown in

Qcond = −Aλ∂T∂n , 33

where n is the normal vector of the boundary surface.

Temperature of heater 3295

290

285

280

T (K

)

275

270

265

2600 100 200 300

Time (s)400 500 600

PresentExperimentFENSAP

Figure 8: Comparison of temperature of heater 3.

NASA

0.6

0.5

0.4

Dro

plet

colle

ctio

n effi

cien

cy

0.3

0.2

0.1

0.0−0

.12

−0.1

0

−0.0

8

−0.0

6

−0.0

4

−0.0

2

0.00

0.02

Curve length (m)

0.04

0.06

0.08

0.10

0.12

Present—mesh aPresent—mesh bPresent—mesh c

Figure 7: Comparison of droplet collection efficiency.

280260240220200180160

Con

vect

ive h

eat t

rans

fer c

oeffi

cien

t (W

/m2 K

)

140120100

80604020

0−0.10 −0.05 0.00

Curve length0.05

LEWICEExperimentPresent—mesh a

Present—mesh bPresent—mesh c

0.10

Figure 6: Comparison of convective heat transfer coefficient.

7 0 0 124006 0 0 124005 0 15500 04 7750 7750 77503 0 15500 02 0 0 124001 0 0 12400

Heater 100 s 10 s 10 sTime

Figure 5: Heating sequence of NASA experiment (power in W/m2).


X

Y

Z

Ice shapet = 100 s

0.05

Ice 100 sWall

0.040.030.020.01

0

0 0.02 0.04X

0.06 0.08 0.1

Y

−0.01−0.02−0.03−0.04−0.05

(a) t = 100 s

Ice shapet = 110 s

X

Y

Z0.05

Ice 110 sWall

0.040.030.020.01

0

0 0.02 0.04X

0.06 0.08 0.1

Y

−0.01−0.02−0.03−0.04−0.05

(b) t = 110 s

Ice shapet = 120 s

X

Y

Z 0.05

Ice 120 sWall

0.040.030.020.01

0

0 0.02 0.04X

0.06 0.08 0.1

Y

−0.01−0.02−0.03−0.04−0.05

(c) t = 120 s

Figure 9: Simulated ice shape following NASA experiment.


3. Solution Procedure

During the solution of the above unsteady conjugate heattransfer model, the water film runback and phase transitionare coupled with the solid heat conduction, due to the factthat the solution of the solid heat conduction provides theinterface heat flux which is needed in the water film energybalance equation; on the other hand, the boundary conditionof the solid heat conduction is provided by the solution of thefilm runback and phase transition. A loosely coupled methodis applied to solve the present model, in which both the waterfilm runback and the heat conduction are iteratively calcu-lated until the heat boundary condition reaches convergence,and during each iteration, the surface temperature T and heatflux Qcond are exchanged, as briefly shown in Figure 2.

Considering that the ice thickness during deicing processis typically controlled at a very small value, the effect of theice shape on air-droplet flow is slight. As a result, the steadyairflow solution is computed as an initial condition and isassumed unchanged during the simulation. Besides, it isaccurate as long as the ice thickness does not exceed a certainlimit due to a protection failure. The RANS equations of air-flow are discretized, using a finite volume method in a secondorder upwind scheme. To simulate the turbulent flow, thetransition SST model, which is shown to obtain good resultsfor wall-bounded flows, such as the airfoil, is utilized. A CFDsolver FLUENT is used to solve the governing equations. Thegoverning equations of droplet flow field are solved by thefinite volume method using the User-Defined Scalar (UDS)transport equation. The droplet volume fraction and velocitycomponents are set as the UDS, and the convective terms,diffusion terms, and source terms are defined by codes whichare programmed using the User-Defined Functions (UDF).The solution of air-droplet flow is the initial condition ofwater film runback and solid heat conduction, and the dataexchange is achieved by interpolation due to the differencebetween flow field mesh and solid mesh.

A loosely coupled method is applied to solve the deicingmodel. At the current time step, both the solid heat conduc-tion and the water film runback are iteratively calculateduntil convergence. During each iteration, the surface temper-ature and heat flux are exchanged, and the boundary condi-tions are updated. The ice shape is obtained by the icingrate or melting rate, and the calculation of the next time stepthen begins. The flowchart is shown in Figure 3, and somebrief introduction is provided below for the flowchart.

(1) Solve the air-droplet flow and transfer the data byinterpolation;

(2) Loop all the surface control volumes and check them.If the input water mass is already known, assume aninitial temperature, solve the mass and energybalance equations, update the value of temperatureand phase state, and provide input conditions foradjacent volumes. Keep the calculation of water filmrunback and phase transition until the calculationsof all control volumes are done; the temperaturedistribution at boundary surface is obtained;

(3) Set Dirichlet boundary condition for solid heatconduction using the temperature distribution ofstep (2) and solve the heat conduction; calculate thedeicing heat flux at boundary.

(4) Update the deicing heat flux of water film energyequation using the data of step (3).

(5) Repeat steps (2)–(4) until convergence, calculate iceaccretion at the moment, and advance to the nexttime step.

4. Results and Discussion

Validations of the present model are conducted, and theresults are compared with the experimental data. Then, asimulation of in-flight deicing process is conducted tooptimize the heating sequence.

4.1. NASA Deicing Experiment in the Lewis Icing ResearchTunnel. In order to perform the in-flight deicing simulation,a deicing experiment model is selected in the very rarerecords available in the open literature, which is conductedin the NASA Lewis Icing Research Tunnel (IRT) byAl-Khalil et al. [25].

The experiment model is a NACA 0012 airfoil with achord of 0.914m (36 in), and the environment conditionsare listed in Table 1. The multilayer structure at the leadingedge protection area is composed of four different layers,which are the erosion shield, the elastomer, the fiberglass,and the silicone foam, with a thickness of 0.2mm, 0.56mm,0.89mm, and 3.4mm, respectively. The structure is shownin Figure 4, and the material properties are listed inTable 2. Seven heating pads are arranged in the elastomerlayer, of which the heating sequence and power density arecontrolled independently. The length is 1.905 cm for heater4; 2.54 cm for heaters 2, 3, 5, and 6; and 3.81 cm for heaters1 and 7. The heating sequence is shown in Figure 5. Heater4 acts as the heat blade, which keeps activated during thewhole cycle to avoid ice accretion on the leading edge. Therear heating pads are activated alternately after 100 s.

The steady air-droplet flow was solved to obtain parame-ters such as the convective heat flux, the shear stress, and thelocal droplet collection efficiency. Then, the data weretransferred to solid mesh by interpolation, and the unsteadydeicing process was coupled solved. Structured grids weregenerated for the solution of air-droplet flow. To verify themesh independence of the solution, three mesh files wereapplied, and the normal distance of the first layer is0.01mm (mesh a), 0.0075mm (mesh b), and 0.005mm(mesh c), respectively. The y+ at the ice protection area ofall three mesh files was controlled around or lower than 1.

Table 3: Environment conditions of in-flight case.

Temperature(K)

Velocity(m/s)

LWC(g/m3)

MVD(μm)

AoA

263 97.6 1 20 0


The simulated convective heat transfer coefficient curvesare shown in Figure 6, which show good agreement with theexperimental data and LEWICE solution. The results of threemesh files match well, indicating that the solutions are meshindependent. The convective heat transfer coefficient reachesthe peak at the stagnation point and drops rapidly as it moves

backwards due to the development of boundary layer. Theexperimental value is slightly larger than that of the simu-lated results, and the turbulence intensity of the icing tunneltest might be a possible explanation. The droplet collectionefficiency was not recorded during the experiment, while itwas simulated by NASA LEWICE code in the literature.

7 0 0 150006 0 0 150005 0 15000 04 10000 10000 100003 0 15000 02 0 0 150001 0 0 15000

Heater Time 75 s 25 s 25 s

Figure 10: Initial heating sequence of in-flight case (power in W/m2).

X

Y

Z

Heat transfercoefficient

2602502402302202102001901801701601501401301201101009080

(a) Convective heat transfer coefficient (W/m2K)

X

Y

Z

Beta0.55

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

(b) Local collection efficiency

Figure 11: Contour of convective heat transfer and droplet impingement.


X

Y

Z

Temperature289288287286285284283282281280279278277276275274273272271

(a) t = 75 s

X

Y

Z

Temperature298296294292290288286284282280278276274272

(b) t = 100 s

X

YTemperature

312310308306304302300298296294292290288286284282280278276274

Z

(c) t = 125 s

Figure 12: Temperature distribution at cross section (K).


X

Y

Z

Ice shapet = 75 s

Ice 75 sWall

0.050.040.030.020.01

0Y

0 0.02 0.04X

0.06 0.08 0.1

−0.01−0.02−0.03−0.04−0.05

(a) t = 75 s

X

Y

Z

Ice shapet = 100 s

0.050.040.030.020.01

0Y

0 0.02 0.04X

0.06 0.08 0.1

−0.01−0.02−0.03−0.04−0.05

Ice 100 sWall

(b) t = 100 s

X

Y

Z

Ice shapet = 125 s

0.050.040.030.020.01

0Y

0 0.02 0.04X

0.06 0.08 0.1

−0.01−0.02−0.03−0.04−0.05

Ice 125 sWall

(c) t = 125 s

Figure 13: Simulated ice shape of in-flight case.


The collection efficiency of the present model and LEWICE isshown in Figure 7, and the results match well. The valuereaches its maximum at the stagnation point, and then thevalue decreases as it moves backwards. There is a dropletshadowed zone when the wrap distance exceeds 0.03m,where the local collection efficiency is zero.

The simulated temperature of heater 3 is shown inFigure 8 with the experimental result and FENSAP simulatedresult provided in [18]. The temperature variation curve ofeach cycle is similar except for the beginning period whenthe solid structure starts to warm up. The result of the presentmodel shows good agreement with the experimental data andFENSAP. The simulated ice shapes at 100 s, 110 s, and 120 s(the moment when heaters are turned on or off) are shownin Figure 9.

At the first stage (0–100 s), only the heat blade isactivated. Due to the solid conduction and the runback liquidwater, the temperature of heater 3 rises. Figure 9(a) showsthat water remains liquid on the surface over the heat blade,and ice forms downstream where the heater is turned off. Therunback ice covers part of the protection area. At the secondstage (100–110 s), heaters 3 and 5 are activated and the tem-perature of heater 3 rapidly rises to 290K. From Figure 9(b),it is observed that the ice over heater 3 melts and the run-back ice area moves backward, due to the fact that theheating enlarges the water runback area. At the third stage(110–120 s), heater 3 is turned off and the temperaturedrops. Figure 9(c) shows that the runback ice, whichforms during the second stage, melts and ice forms overheater 3 as it is turned off.

The simulated temperature of heater 3 is slightly higherthan that of the experimental data. When heater 3 is off, theexperimental temperature is around 270K, and it is about3K lower than the simulated results. At this period, thesurface is under a runback icing condition, which means awater-ice mixed state, and the temperature of such conditionis set at 273.15K (the freezing point) in the thermodynamicmodel during simulation. The possible reasons for the tem-perature difference between experiment and simulation areas follows: (1) The supercooled water film runback phenom-enon, which is observed in the experiments [26–28], has notbeen considered in the present model. In simulation, thetemperature of the runback water film would not be lowerthan the freezing point temperature (273.15K). (2) Therunback water might form beads or rivulet flow and doesnot completely wet the surface. Part of the surface is exposedto air convection, and the temperature is lower than thatwhen the surface is completely wetted. (3) The temperaturemeasuring point in the experiment is beneath the heatingpad, and the precise location is not mentioned in the report;

while in simulation, the average value of the heater area iscalculated as the result. The average value might be largerwhen the heater is turned off, because the temperature atthe margin is higher due to the heating of the adjacentheating pads.

4.2. Deicing Simulation under Different Heating Sequences.The above simulations validate the present model and thesolution method. In this section, another deicing simulationis conducted under in-flight environment conditions to ana-lyze the deicing performance of different heating sequences.The environment conditions are listed in Table 3. Comparedwith NASA icing tunnel experiment, the temperature islower, and the air velocity and liquid water content are larger,which would lead to a severer ice protection condition.

The heating sequence is shown in Figure 10, with a cycleof 125 s. The heat blade (heating pad 4, at the leading edge) iskept activated during the whole cycle; heaters 3 and 5 areactivated during 75–100 s, and other heaters are activatedduring 100–125 s. The ice protection structure model andthe materials are the same with those in Section 4.2.

The convective heat transfer coefficient distribution isshown in Figure 11(a), and the contour of droplet collectionefficiency is shown in Figure 11(b). The convective heattransfer intensity reaches its peak at the stagnation pointand drops rapidly as it moves backwards due to the develop-ment of the boundary layer. Similarly, the droplet collectionefficiency reaches its maximum at the stagnation point, andthe value decreases as it moves backwards. There is nodroplet impingement at the rear part.

Figures 12 and 13 show the solid temperature distribu-tion of the cross section and the ice shapes at t=75 s, 100 s,and 125 s, which are the end time of the three stages. Duringthe first stage (0–75 s), the structure temperature at the lead-ing edge increases due to the activated heat blade, and thetemperature of the heat blade reaches 289K as is shown inFigure 12(a). The temperature of the adjacent area alsoincreases due to heat conduction, while the value is lowerthan that of the heat blade and the heated area is limited,because the thermal conductivity of the structure materialsis low. The water remains at a liquid state over the heat blade,as shown in Figure 13(a), and ice forms downstream. Com-pared with the result of Section 4.2, the ice accretion areaand the ice thickness are larger in this case. All the ice protec-tion area is covered by ice, because the air velocity and liquidwater content are larger and the temperature is lower in thiscase. Later in the second stage (75–100 s), heaters 3 and 5 areturned on, ice starts to melt, and solid temperature increases.At 100 s, ice over the rear part of heaters 3 and 5 meltscompletely, and the surface temperature exceeds 273.15K

7 0 0 150006 0 0 150005 0 15000 04 10000 10000 100003 0 15000 02 0 0 150001 0 0 15000

Heater Time 75 s 30 s 20 s

Figure 14: Altered heating sequence.


as shown in Figures 12(b) and 13(b). While there is ice left inthe front of heaters 3 and 5, the accrete ice thickness is largethere and the power is insufficient for all the accrete ice tomelt. The ice continues to form on the nonheating surfacearea. In the third stage (100–125 s), the temperature atheaters 3 and 5 drops rapidly when the heaters are turnedoff, and runback water freezes over heaters 3 and 5, as shownin Figures 12(c) and 13(c). Ice over heaters 1, 2, 6, and 7melts, and the temperature there increases to a high levelsince the convective heat dissipation there is relatively weak.

At the end of the second heating stage (t=100 s), there isice left unmelted in the front of the surface over heaters 3 and

5, which would lead to ice accretion on that area. Accordingthe discussion above, we know that the deicing power neededin the front is larger than that in the rear part. Therefore, theheating sequence is altered by extending the second stage to30 seconds and shorten the third stage to 20 s, as shown inFigure 14. The total time of a cycle remains the same, whilethe power needed for a cycle is reduced, because the area ofheaters 1, 2, 6, and 7 is larger. The simulated ice shape withthe altered heating sequence is shown in Figure 15.Figure 15(a) shows that after the second stage the surfaceover heaters 3 and 5 is clean with all accrete ice melted.Figure 15(b) shows that after the whole deicing cycle, most

X

Y

Z

Ice shapet = 105 s

t = 105 sWall

0.05

0.04

0.03

0.02

0.01

0Y

0 0.020.01−0.01−0.02 0.040.03X

0.060.05 0.07 0.08

−0.01

−0.02

−0.03

−0.04

−0.05

(a) t = 105 s

X

Y

Z

Ice shapet = 125 s

t = 125 sWall

0.05

0.04

0.03

0.02

0.01

0Y

0 0.020.01−0.01−0.02 0.040.03X

0.060.05 0.07 0.08

−0.01

−0.02

−0.03

−0.04

−0.05

(b) t = 125 s

Figure 15: Simulated ice shape under altered heating sequence.


part of the surface is clean, and only very slight ice accretionoccurs at the location near the heat blade, which would beremoved in the next cycle.

5. Conclusions

Based on the water film runback dynamics and energybalance theory, an unsteady conjugate heat transfer modelfor electrothermal deicing is established, and a looselycoupled solution method is developed. The model is appliedin the simulation of deicing process, and the conclusions areas follows:

(1) In-flight deicing process is very complex due tofactors such as droplet impingement and water filmrunback. The present model is capable of simulatingthe in-flight deicing process. Simulation followingan icing tunnel experiment has been conducted tovalidate the present model, and the results showgood agreement.

(2) The environment conditions would strongly affectthe solid temperature distribution, the water filmrunback, and phase transition. A larger velocity orliquid water content would correspond to a largerrunback icing range, and a higher power of longerheating duration is needed to perform the deicingprocess. Water remains at a liquid state over the heatblade, and ice forms on the surface where the heatingpower is insufficient.

(3) Heating sequence is a key factor for the deicingperformance. A proper heating sequence not onlyleads to a better deicing performance, but also savesenergy. The optimization of heat sequence can beconducted by means of numerical simulation.

(4) However, there are several factors not yet considered,including the ice shedding mechanism, the contactthermal resistance between multilayer materials, andthe anisotropy of material properties.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work was supported by the National Natural ScienceFoundation of China (Grant no. 51206008).

References

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