Ice Accretion Modeling Ice Accretion Modeling ReviewReview

Presented byPresented byProfessor S.A. SherifProfessor S.A. SherifUniversity of FloridaUniversity of Florida

Workshop on Aviation Safety (WAS)Workshop on Aviation Safety (WAS)COPPE/UFRJCOPPE/UFRJ

Rio de Janeiro, BrazilRio de Janeiro, BrazilJune 1, 2010June 1, 2010

Purpose of PresentationPurpose of Presentation

Give the audience a flavor of Give the audience a flavor of some of the engineering some of the engineering calculations involved in calculations involved in calculating how much ice can calculating how much ice can accrete on an aircraft wing for a accrete on an aircraft wing for a given set of flight and weather given set of flight and weather conditionsconditions

When can aircraft icing occurs?When can aircraft icing occurs?

IcingIcing of an aircraft occurs when it of an aircraft occurs when it flies through a cloud of small flies through a cloud of small supercooled water dropletssupercooled water droplets

Two typesTwo types of ice accretion of ice accretion mechanisms have been identified, mechanisms have been identified, resulting in two physically and resulting in two physically and geometrically different formationsgeometrically different formations

Ice Accretion: Type I (Rime Ice)Ice Accretion: Type I (Rime Ice) For low liquid water content, air temperature, For low liquid water content, air temperature,

and flight speed, the accreting ice is and flight speed, the accreting ice is characterized by a characterized by a white opaque colorwhite opaque color and and a a low densitylow density (less than 1 gm/cm(less than 1 gm/cm33).).

This formation is called This formation is called rime icerime ice and is more and is more likely to occur on relatively streamlined likely to occur on relatively streamlined shapes extending into the incoming air.shapes extending into the incoming air.

Rime iceRime ice forms upon impact of the water forms upon impact of the water droplets with the surface and is characterized droplets with the surface and is characterized by a freezing fraction of unity.by a freezing fraction of unity.

Ice Accretion: Type II (Glaze Ice)Ice Accretion: Type II (Glaze Ice) When both the liquid water content and When both the liquid water content and

the flight speed are high, while the air the flight speed are high, while the air temperature is near freezing, the temperature is near freezing, the resulting ice formation will be resulting ice formation will be characterized by a characterized by a clear colorclear color and a and a density near 1 gm/cmdensity near 1 gm/cm33..

This mechanism of formation results in This mechanism of formation results in glaze iceglaze ice which is usually associated which is usually associated with the presence of liquid water and a with the presence of liquid water and a freezing fraction less than one.freezing fraction less than one.

Energy Fluxes on Aircraft WingsEnergy Fluxes on Aircraft Wings

The energy added comprises terms The energy added comprises terms which are due to which are due to freezingfreezing, , aerodynamic aerodynamic heatingheating, , droplet kinetic energydroplet kinetic energy, and , and external sources (such as the deexternal sources (such as the de--icing icing heater).heater).

The energy removed includes terms The energy removed includes terms which are due to which are due to convectionconvection, , evaporationevaporation, , sublimationsublimation, , droplet droplet warmingwarming, and , and aft conductionaft conduction..

Energy Fluxes on Aircraft Wings Energy Fluxes on Aircraft Wings (Cont.)(Cont.)

Both the wing leading edge and Both the wing leading edge and the afterthe after--body regions should be body regions should be considered in any modeling effort.considered in any modeling effort.

The figure illustrates the different The figure illustrates the different modes of energy transfer to and modes of energy transfer to and from an accreting ice surface.from an accreting ice surface.

Dr. S.A. Sherif

Input Parameters Required for Input Parameters Required for ModelingModeling

In order to be able to model icing on a In order to be able to model icing on a wing, we will need knowledge of the wing, we will need knowledge of the following variables following variables a prioria priori::•• altitudealtitude•• flight speed Vflight speed V or the Mach number or the Mach number MM

•• volume median droplet diameter dvolume median droplet diameter ddropdrop•• equilibrium surface temperature tequilibrium surface temperature tss

•• wing configurationwing configuration•• angle of attackangle of attack

Modeling Step No. 1Modeling Step No. 1Compute Freestream Physical Compute Freestream Physical

PropertiesProperties For a given altitude, the freestream For a given altitude, the freestream

physical properties can be evaluated:physical properties can be evaluated:•• pressure ppressure p

•• temperature ttemperature t•• density density ρρ•• kinematic viscosity vkinematic viscosity v•• thermal conductivity kthermal conductivity k

Modeling Step No. 2Modeling Step No. 2Compute VCompute V11/V/V vs. x/Lvs. x/L

Knowing the wing configuration Knowing the wing configuration and angle of attack we can and angle of attack we can determine the ratio of the determine the ratio of the boundary layer edge velocity to the boundary layer edge velocity to the freestream velocity, Vfreestream velocity, V11/V/V, as a , as a function of the nondimensional function of the nondimensional chordwise distance, x/L (as chordwise distance, x/L (as described in Abbott el al. 1945).described in Abbott el al. 1945).

Modeling Step No. 3Modeling Step No. 3Compute pCompute p11 and Tand T11

Knowledge of VKnowledge of V11/V/V enables computing enables computing the pressure and temperature at the the pressure and temperature at the outer edge of the boundary layer:outer edge of the boundary layer:

1

11

21

211

VVMM

pp

1

11

pp

TT

Modeling Step No. 4Modeling Step No. 4Compute pCompute p11 in cases when Cin cases when Cpp is is

givengiven For some surfaces, the coefficient of For some surfaces, the coefficient of

pressure Cpressure CPP along the surface may be along the surface may be available in lieu of the velocity ratio available in lieu of the velocity ratio VV11/V/V. In this case the pressure ratio . In this case the pressure ratio should first be computed using the should first be computed using the following expression:following expression:

PCMpp 21

21

Modeling Step No. 5Modeling Step No. 5Compute VCompute V11

The velocity at the outer edge of the The velocity at the outer edge of the boundary layer should then be found boundary layer should then be found using the equation:using the equation:

1

121

2111

12

ppM

MVV

Modeling Step No. 6Modeling Step No. 6Compute KCompute KT,oT,o

The modified inertia parameter, KThe modified inertia parameter, KT,oT,o, can , can be obtained from the following equation be obtained from the following equation (Bowden et al. 1964):(Bowden et al. 1964):

where the units are knots for Vwhere the units are knots for V∞∞, lb, lbff.s/ft.s/ft22

for for µµ, ft for the chord length, lb, ft for the chord length, lbmm/ft/ft33 for for ρρ∞∞, and ft/s, and ft/s22 for g. The equation gives for g. The equation gives values within values within ±±5% for droplet Reynolds 5% for droplet Reynolds numbers ranging from 25 to 1000.numbers ranging from 25 to 1000.

Ld

gVxK drop

oT 4.0

6.16.07

, 1215.11087.1

Modeling Step No. 7Modeling Step No. 7Compute the Local Collection Compute the Local Collection

EfficiencyEfficiency The local collection efficiency is The local collection efficiency is

computed employing the graphical computed employing the graphical relationships given in several relationships given in several references for a number of airfoils at references for a number of airfoils at different angles (see Brun et al. 1953 different angles (see Brun et al. 1953 for the NACA 65A004 airfoil, and Brun for the NACA 65A004 airfoil, and Brun et al. 1953 for the NACA 65et al. 1953 for the NACA 65--208 and 208 and 6565--212 airfoils)212 airfoils)

Local Collection EfficiencyLocal Collection Efficiency The local collection efficiency, The local collection efficiency, ββ, is , is

defined as the ratio between the locally defined as the ratio between the locally impinging droplet flux and the impinging droplet flux and the freestream droplet flux.freestream droplet flux.

This efficiency is governed by the ratio This efficiency is governed by the ratio of the inertia of the impinging droplets of the inertia of the impinging droplets and their aerodynamic drag.and their aerodynamic drag.

It is primarily a function of the droplet It is primarily a function of the droplet size and distribution, water density and size and distribution, water density and viscosity, freestream velocity, wing viscosity, freestream velocity, wing geometry, and angle of attack.geometry, and angle of attack.

Modeling Step No. 8Modeling Step No. 8Compute mCompute m””ii

The local mass flux impinging on the The local mass flux impinging on the surface may be computed from:surface may be computed from:

WVmi ''

Modeling Step No. 9Modeling Step No. 9Compute hCompute hcc

The heat transfer coefficient, hThe heat transfer coefficient, hcc, at the , at the leading edge can be evaluated using a leading edge can be evaluated using a modified formulation which was modified formulation which was originally introduced for a heated originally introduced for a heated cylinder by Schmidt and Wenner cylinder by Schmidt and Wenner (1943).(1943).

This modified formulation replaces the This modified formulation replaces the cylinder diameter by the airfoil leading cylinder diameter by the airfoil leading edge radius of curvature, d/2.edge radius of curvature, d/2.

Equation for hEquation for hcc

35.35.04.05.0 353072.214.1PrRe

Ls

dL

dLNu LL

Terms in the Heat Transfer Terms in the Heat Transfer Coefficient Equation at the Coefficient Equation at the

Leading EdgeLeading Edge NuNuLL is the Nusselt number based on is the Nusselt number based on

chord length Lchord length L ReReLL is the Reynolds number based on is the Reynolds number based on

chord length L chord length L Pr is the Prandtl numberPr is the Prandtl number s is the distance along the wing surface s is the distance along the wing surface

measured from the leading edgemeasured from the leading edge

Modeling Step No. 10Modeling Step No. 10Compute hCompute hcc in the Aft Region in the Aft Region

(Laminar Regime)(Laminar Regime) For the after region of the wing, two For the after region of the wing, two

possibilities exist depending on the flow possibilities exist depending on the flow regime. For laminar flow, Martinelli et regime. For laminar flow, Martinelli et al. (1943) proposed the following al. (1943) proposed the following equation:equation:

5.05.0

15.0Re286.0

sL

VVNu LL

Modeling Step No. 10Modeling Step No. 10Compute hCompute hcc in the Aft Region in the Aft Region

(Turbulent Regime)(Turbulent Regime)

For turbulent flow, the Nussett number For turbulent flow, the Nussett number expression in the after region may be expression in the after region may be written as:written as:

2.08.0

18.03/1 RePr0296.0

sL

VVNu LL

Modeling Step No. 11Modeling Step No. 11Compute the Air Thermal Compute the Air Thermal

Conductivity and hConductivity and hcc Once the Nusselt number has been Once the Nusselt number has been

computed, the convective heat transfer computed, the convective heat transfer coefficient may be determined for a coefficient may be determined for a given air thermal conductivity. Bowden given air thermal conductivity. Bowden et al. (1964) gave the following for the et al. (1964) gave the following for the thermal conductivity:thermal conductivity:•• T is in T is in RR•• k is in Btu/hr.ft.k is in Btu/hr.ft.FF

8.1/12

5.1

104.2458.1

8.1001533.0

TT

T

k

Modeling Step No. 12Modeling Step No. 12Compute the Relative Heat FactorCompute the Relative Heat Factor The relative heat factor, b, was The relative heat factor, b, was

originally introduced by Tribus (1949) originally introduced by Tribus (1949) and can be expressed as a and can be expressed as a nondimensional quantity of the nondimensional quantity of the impinging flux, the specific heat of liquid impinging flux, the specific heat of liquid water, and the convective heat transfer water, and the convective heat transfer coefficient:coefficient:

c

wi

hcmb

"

Modeling Step No. 13Modeling Step No. 13Compute mCompute m””ff

mm””ff is the mass flux of the fraction of is the mass flux of the fraction of water impinging on the surface and water impinging on the surface and freezing into ice and is given by:freezing into ice and is given by:

where nwhere nff is the freezing fractionis the freezing fraction

""iff mnm

Modeling Step No. 14Modeling Step No. 14Compute qCompute qff

The heat flux to the surface due to The heat flux to the surface due to the freezing of the impinging water the freezing of the impinging water may be computed from:may be computed from:

where twhere tfzfz is the freezing temperature of is the freezing temperature of waterwater

sfzifff ttcmq "

Modeling Step No. 15Modeling Step No. 15Compute Compute ““rr””

Compute the boundary layer recovery Compute the boundary layer recovery factor, r, as given by Hardy (1946):factor, r, as given by Hardy (1946):

where nwhere n11 is is ½½ for laminar boundary for laminar boundary layers and layers and 1/31/3 for turbulent boundary for turbulent boundary layerslayers

1Pr11 2

21 n

VVr

Modeling Step No. 16Modeling Step No. 16Compute qCompute qaeroaero

The heat flux to the surface due to The heat flux to the surface due to aerodynamic heating may be computed aerodynamic heating may be computed from:from:

where J is the mechanical equivalent of where J is the mechanical equivalent of heat 778.26 ft.lbheat 778.26 ft.lbff/Btu and c/Btu and cPP is the is the specific heat at constant pressurespecific heat at constant pressure

P

caero gJc

Vrhq2

21

Modeling Step No. 17Modeling Step No. 17Compute qCompute qdropdrop

Compute the heat flux to the surface Compute the heat flux to the surface due to droplet kinetic energy from:due to droplet kinetic energy from:

gJVmq i

drop 2

2"

Modeling Step No. 18Modeling Step No. 18Compute qCompute qcc

The convective heat flux from the The convective heat flux from the airfoil surface may be computed airfoil surface may be computed from:from:

)( 1tthq scc

Some ObservationsSome Observations

Liquid water present on the wing Liquid water present on the wing surface may be a direct result of water surface may be a direct result of water impinging on the surface or it may be impinging on the surface or it may be due to melting of some of the ice due to melting of some of the ice already in existence.already in existence.

Ice melting may occur due a variety of Ice melting may occur due a variety of reasons such as aerodynamic heating, reasons such as aerodynamic heating, droplet kinetic energy, or the dedroplet kinetic energy, or the de--icing icing heater.heater.

Some Observations (Cont.)Some Observations (Cont.) In order to contain the complexity of In order to contain the complexity of

this model, liquid water present on the this model, liquid water present on the surface will be assumed to result solely surface will be assumed to result solely from direct impinging.from direct impinging.

Since the freezing fraction represents Since the freezing fraction represents the portion of the impinging water the portion of the impinging water freezing into ice, the remaining amount freezing into ice, the remaining amount should represent the portion that should represent the portion that remains as liquid.remains as liquid.

Modeling Step No. 19Modeling Step No. 19Compute the evaporation PotentialCompute the evaporation Potential

The maximum amount of water that The maximum amount of water that can be evaporated (or the can be evaporated (or the evaporation potential) can be evaporation potential) can be computed from the following computed from the following equation according to Sogin (1954):equation according to Sogin (1954):

wv

v

wv

wv

a

v

fa

ve pp

ppp

ppp

MM

TRphm

,1

1,

,1

,1"max,

More on the Evaporation PotentialMore on the Evaporation Potential We assume that the water vapor We assume that the water vapor

behaves like a thermally perfect gas so behaves like a thermally perfect gas so that the thermodynamic properties of that the thermodynamic properties of the vapor can be calculated as though the vapor can be calculated as though the air were not present.the air were not present.

We allow the densities of the water We allow the densities of the water vapor at the surface and the boundary vapor at the surface and the boundary layer edge to be evaluated in terms of layer edge to be evaluated in terms of the partial pressures of the vaporthe partial pressures of the vapor

We account for the influence of induced We account for the influence of induced convectionconvection

Modeling Step No. 20Modeling Step No. 20 Compute the water vapor pressure Compute the water vapor pressure

using empirical correlations. For the using empirical correlations. For the temperature range 492 temperature range 492 TT 672672R, R, Pelton and Willbanks (1972) provided Pelton and Willbanks (1972) provided the following equation:the following equation:

where T is in where T is in R, pR, pv,wv,w is in lbis in lbff/ft/ft22

absoluteabsolute

4525.106.9exp6722117

19.5

, TTp e

wv

Modeling Step No. 21Modeling Step No. 21

Compute the latent heat of Compute the latent heat of vaporization from:vaporization from:

3723 100927.0100839.05696.03.1352 TxTxTe

Modeling Step No. 22Modeling Step No. 22 For a supercooled liquid at a For a supercooled liquid at a

temperature less than 492temperature less than 492R, Dorsey R, Dorsey (1940) provided this correlation:(1940) provided this correlation:

where Awhere A11 = 5.4266514, A= 5.4266514, A22 = = --2005.1, 2005.1, AA33 = 1.3869 = 1.3869 1010--44, A, A44 = 1.1965 = 1.1965 1010--1111, , AA55 = = --4.4 4.4 1010--33, A, A66 = = --5.7148 5.7148 1010--33, , and Aand A77 = 2.937 = 2.937 101055..

4/5

6

2

7

2

48.1

11.374

5

8.17

2

3

21, 10110

8.1

8.1

8.1

3.2exp2117TAATA

wv AT

ATA

TAAp

Modeling Step No. 23Modeling Step No. 23

Compute the mass transfer coefficient, Compute the mass transfer coefficient, hhvv, which may be related to the heat , which may be related to the heat transfer coefficient, htransfer coefficient, hcc, employing the , employing the Lewis analogy:Lewis analogy:

3/23/2

Dc

hLechh

P

c

P

cv

Modeling Step No. 24Modeling Step No. 24 Compute the coefficient of mass Compute the coefficient of mass

diffusion of water vapor in air, D, using diffusion of water vapor in air, D, using the following empirical relationship the following empirical relationship (ASHRAE 2009):(ASHRAE 2009):

where the pressure is in psia, the where the pressure is in psia, the temperature is in temperature is in R, and the diffusion R, and the diffusion coefficient is in ftcoefficient is in ft22/hr./hr.

441

00215.0 5.2

TT

pD

Modeling Step No. 25Modeling Step No. 25

Compute the maximum amount of ice Compute the maximum amount of ice that can be sublimated (or the that can be sublimated (or the sublimation potential) from:sublimation potential) from:

Similar calculations are performed for Similar calculations are performed for sublimation. Many more calculation sublimation. Many more calculation steps are needed.steps are needed.

iv

v

iv

iv

a

v

fa

vs pp

ppp

ppp

MM

TRphm

,1

1,

,1

,1"max,

ConclusionsConclusions

We have presented a summary of We have presented a summary of some engineering calculations used some engineering calculations used in modeling icing on aircraft wingsin modeling icing on aircraft wings

More sophisticated models require More sophisticated models require more intense mathematical analysismore intense mathematical analysis

TimeTime--dependent analysis can be dependent analysis can be modeled using quasimodeled using quasi--steady state steady state calculations under certain conditionscalculations under certain conditions