numerical simulation of heat transfer in mist flow

Numerical Heat Transfer, Part A, 42: 139±152, 2002 Copyright # 2002 Taylor & Francis 1040-7782/02 $12.00+ .00 DOI: 10.1080=10407780290059477

NUMERICAL SIMULATION OF HEAT TRANSFER IN MIST FLOW Giulio Croce Dipartimento di Energetica e Macchine, University of Udine, Udine, Italy

He´ loõ¨se Beaugendre and Wagdi G. Habashi Computational Fluid Dynamics Laboratory, McGill University, Montreal, Canada A numerical simulation of heat transfer over a row of tubes, in the presence of mist flow, is described. Computations include the solution of the flow field around the tubes, the prediction of the motion of water droplets, and the evaluation of the cooling effect of the water film on the tube surface. The entire analysis is carried out using FENSAP-ICE (Finite Element Navier–Stokes Analysis Package for In-flight icing), a simulation system developed by Newmerical Technologies for icing applications. The numerical model is described, including the Navier–Stokes solution, the water thin film computation, the droplet impingement prediction, and the conjugate heat transfer procedure. The predictions are verified against experimental data for different droplet mass flow rates, showing satisfactory agreement and allowing a useful insight in the physical characteristics of the problem.

INTRODUCTION Cold droplets impinging on hot surfaces can provide a signi®cant cooling e ect, mainly due to the evaporative latent heat e ect. This physical mechanism, thus, has been used widely in industry to improve heat transfer rates, to design mist-cooled heat exchangers [1], and for di erent kinds of spray cooling of hot surfaces [2]. Mist-cooled heat exchangers can be used for large power or chemical plants, or small refrigerator radiators. The hot ¯uid ¯ows inside a tube bundle and is cooled by an air stream in a cross ¯ow with water droplets injected in the air stream upstream of the bank of tubes. In a di erent context, the same physical problem is of crucial importance in understanding the behavior of thermal anti-icing devices for aircraft. In such applications, when the aircraft is ¯ying through supercooled clouds, the wing has to be kept at a high enough temperature to prevent ice formation due to a stream of cold outside air and supercooled droplets [3, 4].

Received 3 August 2001; accepted 5 October 2001. Address correspondence to Giulio Croce, Dipartimento di Energetica e Macchine, University of Udine, Via Delle Scienze 208, 33100 Udine, Italy. 139

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NOMENCLATURE CD Cp d D G h H0 k K L mi me v mfin me v N p Pr q Qa Qs Red Sc T

air-droplet drag coe cient speci®c heat droplet diameter tube diameter droplet mass ¯ow rate heat transfer coe cient rotal enthalpy thermal conductivity droplet inertial parameter reference length impinging mass ¯ow evaporating mass ¯ow incoming ®lm mass ¯ow evaporating mass ¯ow ®nite element shape function static pressure Prandtl number heat ¯ux convective heat ¯ux to the air conduction heat ¯ux from the wall droplet Reynolds number schmidt number static temperature

Td Tf Tfin Tr U1 v; vi vl ; vli X; Y x; y a b dij l m Of ; O s r G tij

droplet temperature ®lm temperature incoming ®lm temperature kinetic heat contribution for impinging droplets reference air velocity air velocity vector, air velocity component in i-direction droplet velocity vector, Droplet velocity component in i-direction dimensional spatial coordinates (X parallel to mean ¯ow) x = X=D, y = Y=D nondimensional spatial coordinates (x parallel to mean ¯ow) liquid volume fraction collection e ciency Kronecker delta latent heat of evaporation dynamic viscosity ¯uid and solid domain air density solid±Fluid interface shear stress tensor component

In these examples the heat transfer enhancement is due to the sensible heat contribution of the impinging droplets and to the evaporative latent heat. Empirical correlations and analogies can be used to describe such phenomena, and experimental results can create and add to that information. Computational ¯uid dynamics (CFD), however, is a much more versatile tool that can provide broader range of conditions in a cost-e ective manner. In the present article, an integrated Navier± Stokes system developed for in-¯ight icing analysis [5] is applied to and validated for the evaluation of heat transfer enhancement by mist cooling in an array of cylindrical tubes. This article presents a complete numerical description of the physical situation, shown in Figure 1, avoiding the use of empirical correlations, with all the facets of the problem modeled by the CFD tools. This involves the solution of the ¯ow problem, using a Navier±Stokes code, the evaluation of the collection rate of droplets using an Eulerian-based approach, the computation of the thin ®lm that develops from the front tubes leading edges, the conduction through the tube walls, and the use of a conjugate heat transfer procedure to couple the conductionconvection problems. To validate the complete procedure, computational results will be compared with available experimental data. FLOW FIELD SOLUTION The ¯ow around the tubes has been computed using FENSAP, a threedimensional Navier±Stokes solver whose algorithm is generally described in [6].

SIMULATION OF HEAT TRANSFER IN MIST FLOW

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Figure 1. Physical problem.

Pressure p, velocities vi , and static temperature T are used as independent variables, leading to the following formulation: qr ‡ ( r vi ) ;i = 0 qt

(1)

µ ³

2 q ( r vi ) ‡ ( r vj vi ) ; j = – p;i ‡ m vi; j ‡ vj;i – d ij vk;k 3 qt µ

³

´¶

(2) ;j

2 q( r H 0 – p) ‡ ( r vj H 0 ) ; j = (kT; j ) ; j ‡ m vi vi;j ‡ vj;i – d ij vk;k 3 qt

´¶

(3) ;j

where index i refers to the coordinate direction, r is the density, T the temperature, k the thermal conductivity, and m the dynamic viscosity. Spatial discretization is carried out by a weak Galerkin Finite Element Method (FEM) and the equations are linearized by a Newton method. To advance the solution in time, an implicit scheme is selected, and an iterative GMRES procedure is used to solve the resulting sparse algebraic system. A two-equation k – E model is used to model turbulence. A special element with a logarithmic shape function is used adjacent to walls, allowing the integration of the full equations to the walls through a wall law and avoiding the need for grid over-re®nement in the boundary layer region.

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An accurate and consistent heat ¯ux evaluation is of particular importance when dealing with conjugate ¯uid±solid computations, where the coupling between the ¯uid and solid solvers is only through the exchange of heat ¯ux boundary conditions at the interface. Heat ¯uxes at the walls are postprocessed via the consistent FEM approach of Gresho et al. [7]. In that method, only elements adjacent to a wall are once more reassembled, this time with the temperature imposed as a Dirichlet condition and the heat transfer rate considered as the unknown: Z

NI qdG T = GT

Z

µ

¶

qr H0 qr NI dO ‡ – qt qt O

‡

Z

G

Z

µ

¶

q NI qT ‡ t ij vj dO – r vi H0 ‡ k x xi q q i O

£ ¤ NI ni r vi H0 – t ij vj dG

(4)

where G T is the wall section of domain boundary G , N are the standard FEM shape functions, index I refers to the node and index i to the coordinate direction, ni is ith component of the normal to the boundary, and tij is the shear stress. This approach gives signi®cant gains in accuracy compared with the standard temperature derivative evaluation, in particular for turbulent heat transfer. CONDUCTION SOLUTION The Poisson equation for the solution of the conduction problem in the solid can be considered as a special case of the Navier±Stokes equations with zero velocities. Thus, the same ¯ow code has been reused, in a conduction mode, to evaluate the solid temperature ®eld. Since we are only interested in steady state solutions, we do not need to follow the transient details of the solid. This allows us to obtain faster convergence using a so-called false solid property approach [8, 9]. In fact, we can select an arbitrary solid density (i.e., an arbitrary thermal capacity) in order to force the magnitude of the solid and ¯uid timescales to be comparable. Obviously, the actual value of density does not a ect in any way the steady state solution. CONJUGATE HEAT TRANSFER Thermal coupling between the temperature ®elds within the tube and in the outside ¯ow occurs through the duct wall, thus requiring the conjugate coupling of ¯ow convective and solid conduction heat transfer. Here, the coupling is obtained via an exchange of boundary conditions. This allows us to de®ne the coupling as a purely interfacial algorithm, independent from the details of the solvers in the di erent domains, thus ensuring maximum ¯exibility. Flow solution is initiated with a guessed temperature distribution along the wall; the heat ¯ux computed by the Navier±Stokes ¯ow solver is then used as a boundary condition for the solid conduction problem, which then returns a new temperature pro®le on the wall. The boundary condition choice (Neumann for the solid and Dirichlet for the ¯uid) is made through stability considerations [10].


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The procedure can be summarized as follows: 1. Solve the Navier±Stokes equations ff (H0 ; vi ; p) = 0 in the ¯uid domain O f with Dirichlet boundary conditions at the interface G : » ff (H0 ; vi ; p) = 0 on O f (5) T = Tk on G 2. Evaluate surface heat transfer on G from Eq. (4). 3. Solve the Poisson conduction equations fs (T) = 0 in the solid domain O s with Neumann boundary conditions at the solid=¯uid interface G : ( fs (T) = 0 on O s (6) qT = – Qkss on G qn 4. Evaluate surface temperature TG on the solid=¯uid interface G from the conduction solution. 5. Update the Dirichlet boundary condition on G for the Navier±Stokes solution, introducing a relaxation parameter o : 6. Go to step 1.

T k‡ 1 = (1 – o )T k ‡ o TG

on G

(7)

In the above expression kf and ks represent the ¯uid and solid thermal conductivities, respectively. Such an approach has been widely used in some simpli®ed way in the industrial environments: Usually a few cycles are carried out, using a converged or a nearly converged solution for each problem. Here, a stronger coupling is used, calling the interface procedure at each pseudo time step. This approach corresponds to the Shur complement algorithm for domain decomposition of partial di erential equations described by Funaro et al. in [11] and has proven successful for both antiicing and turbomachinery compressible ¯ow applications [3, 4, 12] and for incompressible ¯ow (where the ¯ow ®eld is decoupled from the thermal one [13, 15]). Strong coupling at each iteration implies that the number of time steps required for the conjugate heat transfer computation is of the same order as the time required for a standalone computation. Actually, if an implicit algorithm is used for the ¯uid and conduction solvers, the global convergence rate is only slightly reduced, in comparison with the standalone computation, as the coupling is not linearized. DROPLET COMPUTATION The collection rate of the droplets on the surface is a function of the air ¯ow ®eld. Although empirical correlations could be used to compute the collection e ciency over simple geometries such as a cylinder [1], it is important to develop a more general computational tool in order to analyze complex geometries, such as staggered multirow tube bundles or anti-icing systems. In the current article, the droplet solution is based on an Eulerian model, which has been described in detail in [16, 17]. Drag, buoyancy, and gravity forces can be taken into account, while it is assumed that no evaporation takes place in the air stream. Both constant or variable diameter sets of droplets, such as a Langmuir distribution, can be considered.

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Using an Eulerian approach, we carry out the computation on the same grid as the ¯ow solution, and we can easily applied it to three-dimensional problems [18]. On the other hand, traditional Lagrangian tracking methods require the tracking of a discrete series of particles from the domain’s inlet to the impinging surfaces, leading to serious di culties in handling 3D complex geometries. If we de®ne the local average liquid volume fraction a and droplet velocity vl , the momentum and continuity equation for the liquid phase can be written as [16, 18]: qa ‡ ( r vli ) ;i = 0 qt

(8)

CD Red q (vli ) ‡ (vlj vli ) ;j = (vi – vli ) 24K qt

(9)

The air drag force term on the right-hand side is a function of the droplets’ Reynolds number Red = r dU1 jv – vl j=m

(10)

K = r d2 U1 =18Lm

(11)

and an inertia parameter

The drag coe cient CD is derived from the well-known empirical drag coe cient for spherical droplets [18] CD = (24=Red )(1 ‡ 0:15 Re0:687 ) d

Red µ 1000

(12)

In the Eulerian context, the water collection e ciency is calculated at every surface point using b = – a vl ¢ n

(13)

where n is the surface normal. FILM COMPUTATION When the droplets impinge on the tube, a thin ®lm of runback water is formed and ¯ows along its surface downstream of the impingement point, driven mainly by wall shear stresses. An accurate evaluation of the heat transfer rate must take into account the energy and mass balance of the impinging water. Again, this problem has been widely studied for icing problems, and is traditionally treated, in a 2D framework, with a simpli®ed control volume approach following the original Messinger [19] model, as shown in [20]. In a 3D Navier±Stokes context we can take into account the ®lm with the introduction of a suitable heat sink at the interface between slat skin and external ¯ow domain.


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The thermal balance of the ®lm yields Qs = Qa ‡ mi Cp (Tf – Td – Tr ) ‡ mev l(Tf ) ‡ mfin Cp (Tf – Tfin )

(14)

where Qs is the conduction ¯ux from the wall, Qa the convective ¯ux to the air, mi the impinging mass ¯ow rate, mev the evaporating mass ¯ow rate, l the evaporation latent heat, Cp the speci®c heat, Tf the local ®lm temperature, Td the impinging droplet temperature, mfin the incoming ®lm mass ¯ow, and Tfin its temperature. The ®ctitious temperature Tr takes into account the kinetic heat contribution of the impinging droplet [20], which can be neglected for low velocity values, such as those encountered in heat exchanger analysis. If a 2D control volume mesh is de®ned at the 3D interface surface, and one assumes that the ®lm ¯ow has the same direction as the wall shear stress vector t, then the incoming mass ¯ow in a cell is de®ned as mfin = –

Z

³ ´ t min mfneigh ¢ n; 0 jt j qO

(15)

where n is the outward normal to the cell boundary and m fneigh is the (scalar) mass ¯ow from the neighboring cells. The cell ®lm mass ¯ow rate mf can then be computed from a mass balance: mf = mi – mev ‡ mfin

(16)

The maximum evaporation mass ¯ow rate is estimated via a heat and mass transfer analogy [21], which can be expressed in the form [21, 20]

mev =

h cpair

³

Pr Sc

´2 3

³ ´ MH2 O ps;f – ps Mair pe – ps;f

(17)

where the convective heat transfer coe cient h comes from the external ¯ow solution, M indicates the molecular weight, cpair is the air speci®c heat, Pr and Sc the Prandtl and Schmidt numbers for air, and the pressure term involves the saturated steam pressure at ®lm temperature ps;f , at external temperature ps and the external pressure pe . Saturation vapor pressure may be evaluated as a function of temperature µ ³ ps (T) = 2337 exp 6879

1 1 – 293:15 T

´

– 5:031 ln

³

T 293:15

´¶

(18)

where the units of ps and T are Pa and K, respectively. Equation (16) yields a system of algebraic equations for the cell mass ¯ow mf . The thermal balance Eq. (14) provides the corrected heat transfer rate to be used as the solid’s thermal boundary condition in the conjugate heat transfer procedure. At a separation point, following physical considerations, one assumes that the water ®lm is shed from the surface.

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Figure 2. G = 10 kg=m2 h (a) streamlines patterns, (b) pressure contours, and (c) temperature contours.

RESULTS A single-row array of tubes will be considered. Tube pitch is set equal to two diameters, the Reynolds number based on the inlet velocity and tube diameter (D = 20 mm) is set equal to 5000, and inlet temperature is 15¯ C. These values are


147

chosen to permit comparison with global experimental results from [1]. To match the experimental set-up, a constant heat ¯ux of 3 kW=m2 is supplied from the tube surface, and the volume mean diameter of the incoming droplets is 65 m m. The air ¯ow is assumed nearly incompressible (inlet velocity equal to 4 m=s). Di erent values of incoming water mass ¯ow G have been considered, from the dry case to the fully wet case. In the following discussion, we assume a coordinate system with X in the streamwise direction and Y in the transverse direction. We neglect the e ect of the droplets on the air velocity and pressure ®eld, as this hypothesis is con®rmed by experimental evidence [1]. Furthermore, as the ¯ow is nearly incompressible, the temperature ®eld is decoupled from the velocity ®eld. Thus, streamlines and pressure contours, given in Figure 2, do not change signi®cantly at di erent incoming water mass ¯ow rates. On the other hand, the temperature ®eld is highly sensitive to the water cooling, as shown in Figure 3. The temperature pro®les in the tube wake at X=D = 1 (X = 0 at the tube center) show the e ectiveness of mist cooling in lowering the peak temperature in the ¯uid.

Figure 3. Temperature pro®le X/D = 1.

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Figure 4. Evaporating mass ¯ow, impinging mass ¯ow, ®lm mass ¯ow: G = 10 kg=m2 h.

Figure 5. Evaporating mass ¯ow, impinging mass ¯ow, ®lm mass ¯ow: G = 100 kg=m2 h.


149

Figure 6. Heat transfer enhancement.

Film balance is depicted in Figures 4 and 5, where impinging mass ¯ow rate, evaporating mass ¯ow, and resulting net ®lm mass ¯owing along the tube are plotted along the streamwise X direction. At very low values of incoming water ¯ow rate G, all the impinging droplets evaporate as soon as they reach the hot surface. Increasing the water mass ¯ow, we observe that some of the water ¯ows downstream, creating a thin ®lm. Thus, in Figure 5, the ®lm mass ¯ow starts from zero at the impingement point (x = – 0:5) and increases as long as the evaporating mass ¯ow rate is less than the impinging one. As the temperature increases along the surface, the evaporating ¯ow increases too, until it become bigger than the impinging one. This take place at x = – :4 for the case G = 10 kg=m2 h and at x = – :1 for the case G = 100 kg=m2 h. Downstream, at low G, the net ®lm ¯ow rate decreases, until the wall temperature is high enough to evaporate all the impinging water, at x = – :28. Beyond this point, the tube surface remains dry. On the other hand, at higher water mass ¯ow G, the net ®lm ¯ux decrease is slower, until the separation point at x = 0:2, where the ®lm is shed away. Since the impinging mass ¯ow rate is zero at x = 0:05, we can expect a reduction in the cooling e ect beyond that point. This is con®rmed by an increase in surface temperature, leading to the corresponding increase in evaporating mass ¯ow shown in Figure 5. Since the most e cient cooling mechanism is due to the latent heat e ect, one needs to maximize the evaporating mass ¯ow rate. Such a maximum is reached for a

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value G = Glim when the wall thin ®lm becomes zero at the separation point. If the droplets ¯ow rate is increased further, the excess water will be shed at this point. Thus, one can expect a strong increase in the average tube heat transfer coe cient in the range 0 < G < Glim , and a less signi®cant enhancement for G > Glim , mainly due to the sensible heat contribution from the impinging water. This e ect is shown in Figure 6 and is in good agreement with experimental data given in [1] for the ®rst tube row of a mist-¯ow heat exchanger. At G > Glim , experimental data show a steeper slope. This is due to some water rivulets driven along the rear part of the tube by gravity, while in the present computation such gravity contributions were neglected. All the above computations have been made assuming a ®xed heat ¯ux on the wall surface. However, with the present procedure the conduction in the tube can also be taken into account. As an example, a computation has been carried out assuming a thin highly conductive copper tube and supplying the heat ¯ux on the inner surface of the tube. The high conductivity allows for transverse conduction in the tube skin, leading to a more uniform pro®le of wall temperature, as shown in Figure 7.

Figure 7. Tube conduction e ect, G = 0 kg=m2 h.


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CONCLUSIONS An integrated procedure for the evaluation of heat transfer in the presence of mist ¯ow has been presented. Droplet and air ¯ow ®eld, as well as tube conduction and evaporating wall ®lm, have been taken into account, and the accuracy of the procedure has been assessed against experimental data. The results show how the proposed numerical approach can be used as a reliable and cost-e ective tool for the analysis and design of thermal devices involving mist cooling. In particular, the numerical approach can be used for quick analysis of several di erent geometrical con®gurations, or even coupled with an optimization algorithm to improve ®nal design, signi®cantly reducing the need for expensive and time-consuming experimental testing. Finally, due to the ¯exibility and generality of a numerical procedure that is fully integrated within a 3D framework, the same procedure can be applied to wide variety of mist ¯ow problems, ranging from heat exchangers to thermal anti-icing devices. REFERENCES 1. Y. Hayashi, A. Takimoto, and O. Matsuda, Heat Transfer from Tubes in Mist Flows, Experim. Heat Transfer, vol. 4, pp. 291±308, 1991. 2. B. W. Webb, M. Queiroz, K. N. Oliphant, and M. P. Bonin, Onset of Dry-Wall Heat Transfer in Low-Mass-Flux Spray Cooling, Experim. Heat Transfer, vol. 5, pp. 33±50, 1992. 3. G. Croce, W. G. Habashi, G. GueÁvremont, and F. Tezok, 3D Thermal Analysis of an Anti-Icing Device Using FENSAP-ICE, AIAA Paper 98-0193, 1998. 4. G. Croce, and W. G. Habashi, Thermal Analysis of Wing and Nacelle Anti-Icing Devices, in G. Comini and B. Sunden (eds.)Computational Analysis of Convection Heat Transfer, pp. 409±432, WIT Press, Southampton, 2000. 5. W. G. Habashi, Y. Bourgault, G. S. Baruzzi, Z. Boutanios, G. Croce, and G. Wagner, FENSAP-ICE: An Integrated CFD Approach to the In Flight Icing Problem, Computational Fluid Dynamics ’98, John Wiley & Sons, vol. 2, pp. 512±517, 1998. 6. G. S. Baruzzi, W. G. Habashi, G. GueÁvremont, and M. M. Hafez, A Second Order Finite Element Method for the Solution of the Transonic Euler and Navier Stokes Equations, Int. J. for Numerical Methods in Fluids, vol. 20, pp. 671±693, 1995. 7. P. M. Gresho, R. L. Lee, R. L. Sani, M. K. Maslanik, and B. E. Eaton, The Consistent Galerkin FEM for Computing Derived Boundary Quantities in Thermal and=or Fluid Problems, Int. J. for Numerical Methods in Fluids, vol. 7, pp. 371±394, 1987. 8. T. C. Hung, S. K. Wang, and F. P. Tsai, Simulations of Passively Enhanced Conjugate Heat Transfer across an Array of Volumetric Heat Sources, Comm. Num. Meth. Eng., vol. 13 no. 19, pp. 855±866, 1997. 9. T. C. Hung, and C. S. Fu, Conjugate Heat Transfer Analysis for the Passive Enhancement of Electronic Cooling through Geometric Modi®cation in a Mixed Convection Domain, Numer. Heat Transfer, Part A, vol. 35 no. 19, pp. 519±535, 1999. 10. M. B. Giles, Stability in Analysis of Numerical Interface Conditions in Fluid-Structure Thermal Analysis, Int. J. for Numerical Methods in Fluids, vol. 25, pp. 421±436, 1997. 11. D. Funaro, A. Quarteroni, and P. Zanolli, An Iterative Procedure with Interface Relaxation for Domain Decomposition Methods, SIAM J. Numer. Anal., vol. 25 no. 6, pp. 1213±1236, 1988.

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12. G. Croce, A Conjugate Heat Transfer Procedure for Gas Turbine Blades, in Heat Transfer in Gas Turbine Systems, R. J. Goldstein (ed.), Annals of N.Y. Academy of Sciences, pp. 273±280, 2001. 13. Y. Chen, M. Fiebig, and N. K. Mitra, Conjugate Heat Transfer of a Finned Oval Tube. Part A: Flow Patterns, Numer. Heat Transfer, Part A, vol. 33 no. 4, pp. 371±385, 1998. 14. L. S. Oliveira, and K. Haghighi, Conjugate Heat and Mass Transfer in Convective Drying of Porous Media, Numer. Heat Transfer, Part A, vol. 34 no. 2, pp. 105±117, 1998. 15. C. H. Cheng, and J. H. Yu, Conjugate Heat Transfer in an Inclined Slab with an Array of Horizontal Channels, Numer. Heat Transfer, Part A, vol. 35 no. 7, pp. 779±796, 1999. 16. Y. Bourgault, W. G. Habashi, J. Dompierre, Z. Boutanios, and W. Di Bartolomeo, An Eulerian Approach to Supercooled Droplets Impingement Calculations, AIAA Paper 97-0176, 1997. 17. Y. Bourgault, H. Beaugendre, W. G. Habashi, C. Y. Lepage, and G. Croce, FENSAPICE: A New Equilibrium Model for Ice Accretion, including Film Runback and Conjugate Heat Transfer, Computational Fluid Dynamics ’98, John Wiley & Sons, vol. 1, Part 2, pp. 723±728, 1998. 18. Y. Bourgault, Z. Boutanios, and W. G. Habashi, Three-Dimensional Eulerian Approach to Droplet Impingement Simulation Using FENSAP-ICE, Part 1: Model, Algorithm, and Validation, AIAA Journal of Aircraft, vol. 37, no. 2, pp. 95±103, 2000. 19. B. L. Messinger, Equilibrium Temperature of an Unheated Icing Surface as a Function of Airspeed, Journal of Aeronautical Sciences, vol. 20, pp. 29±41, 1953. 20. K. Al-Khalil, T. Keith, and K. De Witt, Modelling of Surface Water Behaviour on Ice Protected Aircraft Components, Int. J. Num. Meth. Heat and Fluid Flow, vol. 2, pp. 555± 571, 1992. 21. J. Lienhard, A Heat Transfer Handbook, pp. 526±578, Prentice-Hall, Englewood Cli s, NJ, 1981.