A new icing model has been developed to predict the sponginess (liquid fraction) and growth rate of freshwater ice accretions growing under a surface film of unfrozen water. This model is developed from first principles and does not require experimental sponginess data to tune the model parameters. The model identifies icing conditions that include no accretion, dry accretion, glaze accretion, spongy nonshedding, and spongy shedding regimes. It is a steady state model for a stationary vertical cylinder intercepting horizontally directed spray. The model predicts both the accretion mass growth flux and the accretion sponginess. The model results suggest that spongy shedding and spongy nonshedding regimes are common under the high liquid flux conditions typical of freshwater ship icing. Moreover, the unfrozen liquid incorporated into the spongy ice matrix can substantially increase the ice accretion load over that which would be predicted purely thermodynamically. Despite differences in the experimental setup, the model's performance compares well with two independent freshwater experimental data sets for icing on horizontal rotating cylinders. The model performs well in its prediction of both accretion sponginess and growth rate. The model predicts sponginess with a variation in liquid mass fraction of about 0.2–0.5, over the range of air temperature of 0°C to −30°C, in agreement with observations.
 Icing occurs extensively in a variety of cold environments (hail, aircraft icing, marine icing, and transmission line icing). It requires the presence of a substrate or collecting surface on which liquid aerosol can first gather before solidification and ice accretion can proceed. Beyond these apparent similarities in the icing process, there are also likely to be similarities in the physics of the icing that are less apparent. An example of such a similarity is the phenomenon of spongy ice growth, for which the physics is still not completely understood [List, 1990; Knight, 1991]. However, once spongy solidification and accretion have been understood within the context of one area of icing, that understanding is likely to be of value in other areas of icing as well [Lozowski and Gates, 1991].
 Spongy icing is ice growth that occurs in such a way that it incorporates a portion of the collected spray as liquid into the accretion. The degree of ice sponginess is typically specified by the liquid mass fraction, but it may also be specified by the ice fraction. For a given sample of spongy ice, the sum of the liquid mass fraction (or sponginess) and the ice fraction is unity (see equation (21)).
 Spray icing in the marine boundary layer occurs on vessels and offshore structures, as well as structures and objects along coastlines that are within reach of splashing or wind-borne sprays. Spongy growth has been observed in marine accretions [Makkonen, 1987], but salinity is not a necessary condition for its occurrence. Hail growth and aircraft icing studies for example were the first to bring to light the spongy ice growth problem [Fraser et al., 1953]. Subsequent studies of hail growth have laid the foundations for the current understanding of ice sponginess [Knight, 1968; Lesins, 1983; List, 1963; Morgan and Prodi, 1969; Roos and Pum, 1974].
 Even though marine icing conditions are quite different from those of other forms of atmospheric icing, it is likely that the microphysical processes involved in spongy ice growth are similar. Quite possibly the dendritic growth that is believed to result in the entrapment of surface liquid will be found to be similar under marine and nonmarine conditions.
 A clearer understanding of the physics of spray icing will produce more effective engineering responses to the spray icing problem. The present work is intended to add to the understanding of spray icing, and in particular the phenomenon of spongy ice growth.
 In the following section, we describe the structure of a spongy icing model and its basic assumptions. In the rest of the paper, the liquid film model, the thermodynamics of the falling film and the methods used to predict sponginess and the icing regimes of the model are developed. Next the model performance is compared to data, and its sensitivity to air temperature and liquid water content is examined. Finally, the model's prediction of an icing regime transition is explored. A discussion of the findings and conclusions ends the paper.
2. The Model Structure
 The model has been developed to describe the steady state growth of spray ice on a vertically oriented cylinder. The environmental conditions are considered to be constant. This includes the direction of the wind-borne spray that impinges on the upwind half of the cylinder. The model is an adaptation of and improvement of the model of Blackmore and Lozowski . However, it differs from the latter inasmuch as it is derived from first principles and avoids empirical tuning. It also differs inasmuch as the former model considers a long vertical cylinder, while here, we consider a stubby vertical cylinder whose height and diameter are equal. We do this for the purpose of comparing the model results with horizontal rotating cylinder experiments. We will consider the justification for making such a comparison later. Unlike the model of Blackmore and Lozowski , the present model does not account for the detailed feedback between ice growth, liquid film thickness and heat transfer. However, this simplification in the model allows a consideration of the spongy nonshedding regime and its transition to spongy shedding, something that Blackmore and Lozowski  were unable to do.
Blackmore and Lozowski [1996, Figure 2] show the model's cylindrical configuration, along with horizontally impinging spray and a falling film that is confined to the windward half of the cylinder. By assumption, the falling film moves vertically downward and the model's ice accretion forms on the substrate under the falling film, in an azimuthally uniform fashion on the upwind side of the cylinder. The collected spray is either shed into the airstream from the bottom of the stubby cylinder or accreted by the advancing matrix of ice and entrapped liquid that constitutes spongy ice. There is no inflow of liquid at the top of the cylinder. All of the liquid on the cylinder surface is a result of spray impinging on the vertical surface of the cylinder itself. Figure 1 shows the model's conceptual surface structure with the ice matrix and falling film. In actual shedding, the film thickness would vary with height on the stubby cylinder, increasing from zero at the top to a maximum at the bottom. The model simplifies this picture by considering a film whose thickness is the average of the thicknesses at the top and bottom of the real cylinder. What we have depicted here is the vertically averaged film thickness. The dendritic ice crystal growth occurs as a result of the supercooling of the liquid in the film. The supercooling gives rise to dendritic growth entrapping a portion of the surface liquid. This results in a spongy ice accretion growing beneath a falling film of excess surface liquid.
 The model has four icing regimes which occur successively as the air temperature drops [Lesins, 1983] (1) glaze icing under a slightly supercooled film, (2) spongy ice growth with surface film shedding, (3) spongy ice growth without surface film shedding, and (4) “dry” icing with neither surface liquid nor entrapped liquid.
Blackmore and Lozowski  model the evolution of latent heat within a dendritic layer lying between the icing interface and the spongy ice matrix. In the present model, all of the latent heat of fusion is assumed to evolve at the icing interface. For this purpose, the icing interface is defined as the surface separating the spongy ice matrix and the falling film (see Figure 1). The latent heat of fusion is conducted away from the icing interface initially through the laminar sublayer of the falling film. It is then transported through the outer mixed layer of the film, if it exists. Convective, evaporative, radiative, and sensible heat losses, occurring at the outer surface of the liquid film, finally transfer the heat from the accretion to the cooler environment. It is this heat loss that determines the growth rate of crystalline ice at the icing interface. Under the present assumptions, the modeling of heat transfer is essentially the same as in a conventional icing algorithm with a surficial liquid film [Lozowski et al., 1995].
 The film of falling liquid flows vertically downward adjacent to the icing interface, along the surface of the horizontally advancing ice matrix. The falling liquid film model of Dukler and Bergelin  is used to account for the hydrodynamics of the liquid film. Their model assumes that a laminar layer forms next to the vertical wall (the icing interface in our configuration) with buffer and turbulent layers forming only with sufficiently large film flow. For our purpose, we assume that this outer composite mixed layer, if it exists, offers no resistance to heat transfer. Experience with the model suggests that the film mass flux is rarely large enough to produce a composite mixed layer for typical icing conditions. For example, only laminar falling films were predicted in the model sensitivity studies and the model comparisons with experimental data that are presented below. Nevertheless, in order to preserve model generality, we do allow for the possibility of a mixed layer, which might occur either with very high spray fluxes for the stubby cylinder, or with lesser spray fluxes for a cylinder whose length greatly exceeds its diameter, such as a ship's mast.
 The model describes steady state ice growth with a constant impinging spray flux and constant environmental conditions. For this reason, the ice matrix that is fixed to the substrate, increases in thickness at a constant rate, while the adjacent falling film maintains its thickness. However, the film and the icing interface together, move away from the substrate at the rate of advance of the ice matrix.
3. The Falling Film Submodel
 The Dukler and Bergelin  falling film model is used to determine the liquid film's mass flux and thickness on a vertical surface. The laminar sublayer thickness is important in determining the conductive heat flux through the falling film. The vertical mass flux of the film accounts for the unfrozen surface liquid that is not incorporated into the advancing ice matrix. The falling film submodel is a steady discharge model and it does not account for surface wave propagation, rivulet flow or wind stress effects. There follows a brief description of the falling film model as excerpted from the work of Blackmore .
 The model's falling film moves only vertically, and for simplicity no horizontal component of film velocity is allowed for in the falling film submodel. This allows no surficial liquid flow, for example, from the windward side toward the leeward side of the cylinder, and hence for simplicity we assume that the model accretion is added uniformly to the windward side of the cylinder. The spongy ice mass that is accreted at the icing interface is removed from the falling film as the liquid in the film flows over the icing interface. The model neglects the possible circulation or flow of liquid within the structure of the accreted spongy ice.
 Unlike the model of Blackmore , the present model uses a “no icing” falling film to approximate the falling film in actual spray icing. In other words, the present approach does not allow for an adjustment in the film thickness to account for the ice and liquid incorporated into the advancing ice matrix. This simplification makes the thermodynamic and sponginess modeling easier to tackle and the model's performance less difficult to understand. Under the conditions considered in this paper, the maximum film thickness, which occurs at the bottom of the cylinder, ranges up to about 200 μm.
 The conservation of water substance for the laminar layer (Figure 2) is expressed by:
where R4 is the impinging spray flux, I0 represents the formation rate of pure ice that constitutes the solid portion of the ice matrix (kg m−2 s−1), R0 is the flux of water entrapped by the ice matrix, and Rt is the mass flux of liquid shed from the cylinder in the film flow, given by:
where R2 is the mass flux shed in the laminar layer, and R3 is the mass flux shed in the mixed layer of the falling film. If only a laminar layer exists, Rt = R2 then as shown in Figure 2. We note here that we have retained, where feasible, the notation of Blackmore and Lozowski .
 In the model, we take the impinging spray flux, R4, to be a known environmental input parameter. The thermodynamics of the layers of the falling film, along with the model's icing regimes (for details, see section 6) are used to calculate the total accretion flux, (I0 + R0). Once the total accretion flux and the impinging spray flux are known, equation (1) is used to calculate Rt, the total liquid flux. Blackmore and Lozowski  have provided the details of how the layer thicknesses and fluxes in the falling film (i.e., R2, R3 of equation (2)) are determined.
4. The Thermodynamics of the Falling Film
 In the model, we assume that only conductive heat transfer occurs in the laminar layer of the falling film. The mixed layer, if it exists, is modeled as offering essentially no resistance to the heat transfer between the outer surface of the laminar layer and the environment. Hence the mixed layer is taken to be isothermal, at the temperature of the outer surface of the laminar layer. Under these physical assumptions, the heat balance for the outer surface of the laminar layer is (cf. Figure 4) [Blackmore and Lozowski, 1998]:
where qc, is the conductive heat flux that originates at the icing interface, which is directed through the laminar layer into the mixed layer of the film. The other terms are common in the spray icing literature and are usually present in conventional icing models [Makkonen, 2000]. They are qa, the convective heat flux to the airstream, qe, the evaporative heat flux to the airstream, qr, the net radiative heat flux to the airstream, and qs, the flux of sensible heat required to warm the impinging spray collected by the film. The heat fluxes at the outer surface of the laminar layer are considered positive if heat flows toward the outer surface and negative if heat flows away from the outer surface. The conductive flux, qc, is given by:
where kw, is the thermal conductivity of water (W m−1 K−1), δ, is the thickness of the laminar layer (m), and ΔT is the supercooling (considered positive) at the outer surface of the laminar layer (K). We assume that the icing interface (see Figure 3) is at the equilibrium freezing temperature (i.e., 0°C).
Equation (3) is used in the model to calculate the temperature and supercooling, ΔT, at the outer surface of the laminar layer, which is also the temperature of the mixed layer, if one exists. Once this temperature has been determined, the following equation describing the heat balance of the laminar layer can be used to solve for I0, the formation rate of pure ice at the icing interface. The heat balance for the laminar layer is (see Figure 3):
where qlf, is the flux of latent heat evolved at the icing interface and conducted into the laminar layer. The next term, qsl, is the flux (considered positive) of sensible heat required to warm the impinging water entering the laminar layer either from the mixed layer, or directly from the spray if there is no mixed layer. This sensible heat flux is the product of the specific heat of water, the mass flux of water entering the laminar layer, and the temperature difference between the incorporated water and the water film. The curvature of the water film is ignored in the calculation of the mean water film temperature and the sensible heat flux. Because the flux of latent heat, qlf, may be expressed as a product of the formation rate of pure ice, I0, and the latent heat of fusion Lf (i.e., qlf = I0Lf), and because qc is known from equation (4), equation (5) can be used to solve for I0.
 So far we have described model development that essentially parallels that of a conventional icing algorithm with a surficial liquid film [Lozowski et al., 1995]. Like conventional icing models, this allows for calculation of the pure ice growth rate (represented by I0) but not for the determination of the flux of liquid entrapped in the ice matrix, R0. In order to calculate this flux, we will use a theoretical cum empirical description of spongy liquid inclusion at the icing interface. This is described in the following section.
5. The Prediction of Sponginess
 In crystal growth theory, it has long been recognized that the linear growth rate of the tips of dendrites is a function of the supercooling in the bulk water into which the dendrite is growing [Chalmers, 1964]. This concept has been used in atmospheric icing research for some time to help explain the growth mechanism of water-laden spongy ice [Knight, 1968; Makkonen, 1987; List, 1990]. These authors reason that, at the icing surface, dendritic growth occurs that is driven by the supercooling of the liquid film. In fact, it seems plausible that as the supercooling in the liquid film increases, more rapid crystal growth results, with an effect on the liquid entrapment.
 In order to develop a model for the prediction of accretion sponginess, we begin with the definition of ice fraction, f:
where I0 represents the formation rate of pure ice at the icing interface. This formation of ice constitutes the crystalline ice matrix, while R0 is the mass flux of liquid water incorporated into this ice matrix. The ice fraction can also be expressed in terms of the ratio of two growth velocities as follows:
where the first velocity, Vi, is the rate of advance of an icing interface assuming negligible sponginess (i.e., Vi = I0/ρi), and the second velocity, V1, is the rate of advance of the ice dendrite tips at the icing interface, relative to the liquid in the vicinity of the tip (i.e., V1 = (I0 + R0)/ρw). The quantities ρi and ρw are the densities of ice and water, respectively.
 First we derive an expression for Vi, the rate of advance of an icing interface with no liquid entrapment. In equation (5), the flux of sensible heat required to warm the water entering the laminar layer, qsl, is typically a term that is small compared with qc and qlf, and it can therefore be neglected with little error. The result is that the latent heat term, qlf, is approximately equal to the conductive term, qc. This leads to a simplified equation for the flux of pure ice:
where Lf is the latent heat of fusion of water at 0°C. Using equation (8), an approximate expression for Vi may be written:
where qc, the conductive heat flux, is given by equation (4). Substituting for qc from equation (4), the final form for Vi is:
 An expression for V1, the rate of advance of the ice dendrites, will now be developed and substituted into equation (11). We begin with the empirical crystal growth equation:
where V is the rate of advance of a single ice crystal into bulk supercooled liquid with a uniform temperature far from the ice crystal, and ΔTi is the bulk liquid supercooling far from the dendrite. The constants a and b have been evaluated experimentally by, for example, Tirmizi and Gill  (a = 1.87 × 10−4, b = 2.09) and by Kallungal and Barduhn  (a = 1.18 × 10−4, b = 2.17) for ice dendrite growth. Average values for the coefficient and exponent were calculated based on these studies and are used in the present model equation for dendrite growth (i.e., a = 1.53 × 10−4, b = 2.13). Equation (12) assumes a shape-preserving dendrite which has a radius of curvature at its tip that is unchanging with time.
 Next, we use this expression for V, the rate of advance of a single freely growing dendrite to model V1, the rate of advance of the dendritic front relative to the liquid in the vicinity of the dendrite tips in spongy icing. We replace ΔTi, the supercooling far from the idealized freely growing dendrite, with the supercooling at the outer surface of the laminar layer in spongy icing (i.e., ΔTi = ΔT). This is at once a naïve and simplifying assumption. It is perhaps reasonable to expect some functional relation between the two supercoolings, and we have chosen to represent this as an equivalence, rather than by introducing unknown but tunable parameters into the model. In any event, an uncertainty in ΔT of a factor of 2 gives rise to an uncertainty in ice fraction of less than 10% (equation (21)). Under these modeling assumptions, the equation for the rate of advance of the ice dendrites in spongy growth is:
where R/δ is the ratio of the radius of curvature of the dendrite tip to the thickness of the falling film. Next, we derive an expression for this ratio, based on the assumption that the dendrite tip in spongy growth is at or near equilibrium with the liquid in the vicinity of the tip. Figure 4 shows a two-dimensional schematic cross section of dendrite tips that protrude into the falling film from the icing interface. At the tips of the dendrites in spongy icing, we assume that the Gibbs–Thompson equation holds, and that the supercooling in the liquid at the dendrite tip is related to the radius of curvature at the tip according to:
where ΔTr is the supercooling at the dendrite tip, Γw is the Gibbs–Thompson coefficient and κ is the curvature at the tip of an idealized two-dimensional dendrite (κ = 1/R). This is an idealization of three-dimensional dendrites that are likely in spongy growth. The curvature of a three-dimensional dendritic tip is κ = 1/R + 1/R′, where R′ is the larger radius of curvature at the tip in the basal plain. Since R′/R is approximately 50 in freely growing ice dendrites [Tirmizi and Gill, 1987; Furukawa and Shimada, 1993], we estimate the effective curvature in our two-dimensional idealization as κ = 1/R. (Values of R under the conditions considered in this paper range from about 0.5 to about 5 μm.) The Gibbs–Thompson coefficient is given by:
where γw is the interfacial surface tension between the liquid and solid phases and Δsf is the entropy of fusion per unit volume. The entropy of fusion per unit volume is Δsf = ρwLf/Tf where Tf is the equilibrium freezing temperature (Tf = 273.16 K). The value of γw, the interfacial surface tension, used in the model, is an average value based on the reported values of Kallungal and Barduhn , Jones , Skapski et al. , and Poisot  (i.e., γw = 4.75 × 10−2 J m−2). Since it may be shown that the model's ice fraction is linearly related to γw, any uncertainty in γw will be directly reflected in the ice fraction. Using this value, the Gibbs–Thompson coefficient is calculated to be Γw = 3.885 × 10−8 mK.
 Since we assume that the conductive heat flux through the laminar falling film is constant (i.e., no heat source in the film), the temperature and hence the supercooling are linear functions of the distance perpendicular to the icing interface in the falling film. The supercooling at the icing interface is assumed to be negligible, with a maximum value at the outer surface of the film. The temperature as a function of displacement is:
where y is defined as the linear distance perpendicular to the icing interface (Figure 3), ΔT = ΔT(δ), and ΔT/δ is the gradient of supercooling in the laminar layer. If the dendrite tip protrudes a distance, yR, into the laminar layer, then using equation (19), and assuming that the isotherms are parallel to the interface, the supercooling at the dendrite tip is:
Hence the ratio of dendritic penetration to the thickness of the laminar layer is equal to the ratio of supercoolings (i.e., yR/δ = ΔTR/ΔT).
 Even though the cross-sectional shape of ice dendrites in spongy ice growth is not known, we have been proceeding on the assumption that the growth is shape-preserving much like freely growing dendrites. Solutions for the growth rate of freely growing dendrites have been derived for many different tip geometries and growth assumptions [Ivantsov, 1947; Horvay and Cahn, 1961]. Tirmizi and Gill  and Furukawa and Shimada  have observed freely growing ice dendrites that are elliptical paraboloids in the vicinity of the tip. Faced with the difficulty of having no published observations of the shape of dendrites in spongy icing, we assume the two-dimensional equivalent to the hemispherically tipped needle dendrite. This morphological assumption leads to a semicircular dendritic tip penetrating the falling film at the icing interface as shown in Figure 4. The array of dendrites, because of their tip geometry, penetrate the laminar layer only to a depth, yR, which equals the radius of curvature at their tips (i.e., yR = R). With this morphological assumption in place, we can now restate the ratio equality that follows equation (20) as R/δ = yR/δ = ΔTR/ΔT. Next, we combine equations (13) and (17) to get R = Γw/ΔTR = Ci/ΔT or ΔTR/ΔT = Γw/Ci. Thus the ratios ΔTR/ΔT, Γw/Ci, and R/δ are equal with a value of 0.0442. The value of this ratio is substituted into equation (16) to give the equations used in the model for the calculation of the sponginess, λ, and the ice fraction, f:
where the value of the constant C is 0.56.
Equation (21) is used in the model after the supercooling at the outer surface of the film has been determined using the heat balance at the outer surface of the laminar layer (equation (3)). Once ΔT and f are known, along with I0, the formation rate of pure ice, the entrapment flux of liquid, R0, may be calculated as described in the next section.
6. The Icing Regimes
6.1. The “Dry” Icing Regime
 The model uses the heat balance equation for the mixed layer (equation (3)) to calculate the temperature at the outer surface of the laminar layer, T2. If it finds that T2 ≥ 0°C, the model predicts that no icing will occur. This can occur if the spray is above freezing as in marine icing. Conversely, if it finds that T2 < 0°C, the model predicts ice growth. The formation rate of pure ice, I0, is calculated using the heat balance equation for the laminar layer (equation (5)). If I0, is found to exceed the flux of impinging spray (i.e., I0 ≥ R4) then “dry” icing is predicted by the model, no unfrozen liquid is either shed or accreted, and the ice growth rate is determined by R4.
6.2. The Spongy Shedding Icing Regime
 The spongy shedding icing regime is predicted in the model if the impinging mass flux of water, R4, is large enough to form both a spongy accretion and a falling film of excess liquid. This regime occurs when the model predicts that I0 < R4. The accretion sponginess, λ, is calculated from equation (21). Thereafter, the entrapped liquid flux for the spongy shedding growth regime is calculated from:
6.3. The Glaze Icing and Spongy Nonshedding Regimes
 The glaze icing regime is predicted by the model if the impinging spray forms an accretion with no sponginess (i.e., λ = 0) and there is a falling film of excess liquid. This occurs in the model, if the accretion sponginess calculated in equation (21) is found to be λ ≤ 0. In this regime, λ is set to zero and the entrapped liquid, R0, is also set to zero.
 The spongy nonshedding regime is predicted by the model if the impinging spray flux, R4, is large enough to form a spongy accretion but not a falling film of excess liquid. In this regime, the model predicts I0 < R4, a positive value of λ (from equation (21)) and a value of R0 from equation (22). In the spongy nonshedding regime, I0 + R0 > R4, and hence there is no excess liquid available to form a falling film. For this regime the final value of R0 is calculated with:
In the case of the model's spongy nonshedding ice growth regime, the accretion sponginess will decrease from its value calculated with equation (21), and is instead computed from:
where R4 = I0 + R0. In this event, the flux of shed liquid, Rt = 0. The spongy nonshedding regime can be thought of as icing with a net collection efficiency of 1.0 (i.e., no liquid shedding), and a thermodynamically determined sponginess. We note here that in the present model, we do not consider any loss of impinging spray due to splashing.
7. Experimental Comparisons
 The performance of the model has been examined by comparing its prediction of sponginess and icing flux with measurements made in the Marine Icing Wind Tunnel at the University of Alberta (J. Shi, unpublished data, 1996). These freshwater icing experiments were performed with a slowly rotating cylinder of diameter 4.0 cm. The cylinder was oriented horizontally below the spray nozzles. The nozzles were set to generate two spray mass fluxes (0.05 and 0.1 kg m−2 s−1) at the icing cylinder. Nineteen icing experiments were completed at the lower spray flux (0.05 kg m−2 s−1) and 39 experiments were completed at the higher flux (0.1 kg m−2 s−1). The low spray flux data set was obtained at a single airflow speed of 10 m s−1, while the high flux data were obtained at airspeeds of 10, 20, and 30 m s−1. The airflow and gravity accelerated the spray vertically downward toward the cylinder, cooling the droplets on the way. Experiments were run for air temperatures ranging from −0.5°C to −25°C.
 The average icing flux was determined for each experiment. The sponginess of the accreted ice was also determined by first removing the accretion and then subjecting the samples to a calorimetric measurement of sponginess. The uncertainty in measuring sponginess or liquid mass fraction was estimated to be ±0.1.
 Comparison of these data for a horizontal, slowly rotating cylinder in the experiments with a vertical nonrotating cylinder in the model is not ideal. We do not make a more suitable comparison because we are unaware of any experimental sponginess data for the vertical stubby cylinder configuration of the model. At the same time, we have avoided trying to model the horizontal rotating cylinder configuration of the experiments, because of the complexity of the wind-driven surface liquid flow. We nevertheless proceed with the comparison because there are some similarities between the two configurations. For example, similar cylinder diameters and environmental conditions should give rise to a similar heat transfer regime. Also, similar spray fluxes should produce similar liquid film thicknesses. Consequently, we will proceed with a comparison with the Shi data and also, later, with the data of Lesins et al.  for a horizontal rotating cylinder in the University of Toronto Icing Wind Tunnel. However, before making a comparison with Lesins et al. , we will compare sponginess, λ, and icing flux, (I0 + R0), with the Shi data from the University of Alberta Marine Icing Wind Tunnel in Figures 5 and 6.
 We will begin with a consideration of spray impingement temperature, which is an essential input variable for the model. To the best of our knowledge, the spray impingement temperature has never been directly measured in any icing experiments. It is difficult even to imagine how one might go about measuring it. Consequently, one is left with trying to infer the impinging droplet temperatures by other means such as a single droplet trajectory model with heat transfer. Various authors have devised such models [e.g., Lozowski et al., 1979; Gates et al., 1988]. They are instructive but not definitive. They demonstrate, for example, that the final droplet impingement temperature depends, inter alia, on the initial droplet temperature and injection velocity, both of which are typically unknown. It also depends on droplet size, air temperature, airspeed, the distance from the sprayers to the measuring section and the geometrical details of the wind tunnel contraction. These studies have shown, not unexpectedly, that small liquid droplets (∼20 μm diameter) quickly achieve thermal equilibrium with the airstream, while larger droplets (∼100 μm diameter) do not achieve thermal equilibrium, and can have impingement temperatures many degrees warmer than the air temperature. In addition, the spray may cool the air, although it is readily shown that for spray liquid water contents up to about 10 g m−3, the air temperature change is almost two orders of magnitude smaller than the spray temperature change. So, without knowing the details of the droplet spectrum, the injection conditions and the wind tunnel configuration, it is difficult to make a reliable estimate of spray impingement temperature using such models. In view of this dilemma, we have chosen here to use a simple estimate of spray impingement temperature, equal to the average of the freezing temperature and the air temperature.
 Since the spray impingement temperature was not measured (J. Shi, unpublished data, 1996), we estimate the droplet temperature as the mean of 0°C (approximately the nozzle exit temperature) and the air stream temperature. This means that our model predictions are dependent on the assumption of partially supercooled impinging spray. In addition, we use the cylinder diameter from the experiment as both the diameter and height of the cylinder in the model predictions.
 In Figure 5, for the 19 trials with low spray flux (0.05 kg m−2 s−1), the model-predicted sponginess compares well with the experimental data, given the experimental error bars of ±0.1 The model predictions and experimental observations of sponginess both have a range of from 0.2 to 0.4, and there appears to be little systematic error in the predictions. The model-predicted icing fluxes also compare quite well with the experiments. The largest absolute error is less than 0.005 kg m−2 s−1 for the range of icing fluxes shown in Figure 6.
 In Figure 7, for the 39 trials with high spray flux (0.1 kg m−2 s−1), the model-predicted sponginess compares well once again with the experimental data. The model predictions of sponginess lie within ±0.1 of the experimental measurements of sponginess, consistent with the experimental error bars. The model predictions and experimental observations of sponginess both have a range from about 0.2 to about 0.45, and there appears to be little systematic error in the predictions. The model-predicted icing fluxes also compare quite well with the experiments. The largest absolute error is about 0.01 kg m−2 s−1 for the range of icing fluxes shown in Figure 8. There is a systematic overprediction by the model at lower icing fluxes. Next, we will compare the model prediction of sponginess with the experimental data of Lesins et al. . We begin by briefly describing their experiment.
Lesins et al.  performed freshwater icing experiments in a refrigerated wind tunnel using a slowly rotating (0.5 Hz) cylinder of diameter 1.9 cm. The cylinder was oriented horizontally 1.8 m above the spray nozzles. An upward airflow with a speed of 18 m s−1 carried the spray up to the cylinder, cooling the droplets en route. They varied the liquid water content from 2 to 40 g m−3 while the air temperature was varied from −2°C to −20°C. They then measured the thickness of the accretion on the cylinder, along with the ice fraction of a sample of the accretion using a centrifuge method.
Lesins et al.  performed a least squares best fit to their data that resulted in an empirical equation for ice fraction, f, as a function of liquid water content. They averaged their data over an air temperature range from −4°C to −16°C, because the measured ice fractions over this temperature range had a variation that was less than the uncertainty in their measurements (about ±0.1). Their statistical fitting function yields an asymptote at a solid fraction of f = 0.25, even though their lowest solid fraction measurement was 0.33. Their fitted curve for ice fraction as a function of liquid water content is shown in Figure 9 as a solid line, with error bars that express an uncertainty of ±0.1.
 Since the spray temperature on impingement was not measured by Lesins, we have used the mean of 0°C and the air temperature as an estimate of the temperature of the impinging spray. Once again, the cylinder diameter and height in the model is taken to be equal to the cylinder diameter in the Lesins et al.  experiment (1.9 cm). Since the empiricism from the work of Lesins et al.  is valid for an air temperature range from −4°C to −16°C, we have shown model ice fraction curves for air temperatures of −4°C (square symbols), −10°C (circular symbols), and −16°C (diamond-shaped symbols) in Figure 9. In Figure 9, the model-predicted ice fraction compares well qualitatively with the result of Lesins et al. . The model's prediction shows a tendency similar to the Lesins et al.  data over the full range of liquid water content, and it lies within the experimental error of the results of Lesins et al.  over a significant portion of the range of liquid water content. Nevertheless, it is clear that the model overpredicts ice fraction at high liquid water content and underpredicts ice fraction at low liquid water contents. The close proximity of the three model-predicted curves agrees qualitatively with the observation of Lesins et al.  that the ice fraction (measured with an uncertainty of ±0.1) is nearly constant over the air temperature range −4°C to −16°C.
 Over the range of liquid water content shown in Figure 9, the model predicts three different icing regimes as outlined in section 6 above. When the ice fraction, f = 1.0, the model predicts a dry icing regime. The second, third and fourth square and circular symbols, and the second through eighth diamond symbols from the left, at low liquid contents, identify the spongy nonshedding ice growth regime. In this region, the curves have a steep slope, suggesting that that the range of liquid water contents that produces the spongy nonshedding icing regime is quite small. A direct comparison with the work of Lesins et al.  cannot be made concerning this point. However, it is clear that the model slopes are much higher in this regime than the smoothed data of Lesins et al.  would suggest.
 With increasing liquid water content, the model predicts a transition from the spongy nonshedding regime to the spongy shedding regime. At higher liquid water contents, the curves are close to horizontal, suggesting an insensitivity of sponginess to liquid water content in this regime. In summary, the model predicts three different icing regimes for the range of liquid water contents shown in Figure 9. The transition from the spongy nonshedding regime to the spongy shedding regime will be discussed further in section 9. Next we examine the sensitivity of the model to variations in air temperature.
8. The Model Sensitivity
 In order to examine the model's performance over a range of air temperature, we selected input icing conditions similar to those of the high spray flux experiment (J. Shi, unpublished data, 1996) described in section 7 above. This selection allowed us to plot sponginess for the 14 experimental runs with airspeeds of 10 m s−1 in Figure 10. We also compare icing flux in Figure 11. The input airspeed is 10 m s−1 with a spray flux of 0.1 kg m−2 s−1 (i.e., a liquid water content of 10 g m−3). Once again, the spray impingement temperature is taken to be the mean of 0°C and the air temperature.
 In Figure 10, the model-predicted curves (solid lines) are for three airspeeds 5, 10, and 30 m s−1. In all three curves, the onset of icing begins very near the freezing point in the spongy shedding regime, and the sponginess increases slowly as the air temperature decreases. There is no predicted glaze ice regime under these conditions. The sponginess at the onset of icing is a function of the airspeed.
 As the air temperature decreases, the sponginess in all three model-predicted curves increases until reaching the transition point from the spongy shedding regime to the spongy nonshedding regime. This transition point is marked by the discontinuity in each of the curves. These transition points occur at lower air temperatures as the airspeed increases, and each sponginess curve becomes nearly linear after the transition point.
 As the air temperature decreases in the spongy nonshedding regime, all the collected spray flux is either solidified or remains as entrapped liquid according to the heat balance at the icing surface. Therefore, with a further decrease in air temperature, the accretion finally reaches zero sponginess or dry icing. In Figure 10, this transition occurs where the model-predicted curves intersect the air temperature axis (i.e., at zero sponginess). This occurs at air temperatures of about −40°C and −70°C for the 30 and 10 m s−1 model-predicted curves, respectively. Thus the model predicts a third kind of transition for the range of conditions represented in Figure 10, namely the transition from spongy nonshedding to dry icing.
Figure 11 shows the onset of icing near the freezing point and the transition from spongy shedding to spongy nonshedding icing. The spongy shedding regime begins at an air temperature near the freezing point and exhibits a nearly linear variation of icing flux with air temperature. This nearly linear form of the icing flux function is in part caused by the fact that the sponginess is relatively insensitive to air temperature. This relative insensitivity of sponginess, which is apparent in Figure 10, was first proposed by Makkonen  for sea spray icing, and subsequently by Makkonen  for freshwater icing. It is also clear that the sponginess values in Figure 10 could lead to a considerable increase in the accretion loading over what would be predicted were sponginess ignored.
 In Figure 10, the model-predicted sponginess curve that is marked with solid circles compares reasonably well with the experimental data (open circles). The model underpredicts sponginess over the air temperature range −20°C to 0°C. However, the qualitative form of the experimental data appears to match the model-predicted curves quite well, considering the uncertainty of ±0.1 in the experimental measurements. In Figure 11, the model-predicted curve for icing flux that is also marked with solid circles, compares reasonably well with the experimental points (open circles). The model slightly underpredicts the icing flux in the air temperature range 0°C to −10°C and then overpredicts at lower air temperatures.
9. The Spongy Icing Transition
List  proposed that the ice accretion in spongy icing consists of a dendritic mesh growing into a surface water skin that may be up to 1 mm in thickness. He suggests that the physical picture of the water film includes a nearly linear temperature gradient that is responsible for conductive heat loss from the advancing dendrites. Knight  argues that “if an ice dendrite or plate were to extend a little farther into the skin than its neighbors, it would see a lower temperature, grow faster than its neighbors, see therefore a yet lower temperature, grow yet faster, and so on.” He goes on to suggest that such an unstable dendritic growth would occur throughout the surface water film. In this way, the liquid film would be filled with dendrites leaving little or no surface liquid.
 The physical structure of spongy icing that List  proposes is quite similar to the surface structure of our spongy shedding regime. We suggest that the spongy shedding regime is the result of stable and steady dendritic growth that occurs over a range of environmental conditions. Hence, we disagree with Knight's conceptual model, at least in general. However, the physical picture that Knight  describes is similar to the surface structure of our spongy nonshedding regime. In our spongy nonshedding regime, the dendritic growth has a rate of advance that is sufficient to entrap all of the collected spray flux in the advancing dendritic mesh. This occurs when the imposed heat loss to the environment produces dendritic growth that is fast enough to allow all of the dendrites to reach the surface of the liquid film.
Blackmore and Lozowski  model the spongy shedding regime of spray icing with an ad hoc empirical approach. In the present approach we model both the spongy shedding regime and the spongy nonshedding regime from first principles of the growth of an array of dendrites at the icing interface. This new approach allows for the prediction of the transition between the two regimes. This transition is evidenced by the discontinuities in the model-predicted curves in Figures 9, 10, and 11. Therefore, the model's transition is a function of liquid water content and air temperature, both of which influence the icing surface thermodynamics.
 Each of the spongy growth transitions from spongy nonshedding to spongy shedding, predicted by the model, occurs when the rate of advance of the model's dendrites matches the rate of accretion of liquid water at the icing surface. For the three transitions in Figure 9, these rates are 3.7, 7.55, and 12.3 cm h−1, at −4°C, −10°C and −16°C, respectively. In the model, this occurs at laminar film thicknesses of 32.9, 41.7, and 49.1 μm, respectively. Even though Blackmore and Lozowski  could not calculate these film thicknesses with their model, they hypothesized that the surface liquid film would be completely entrapped by the advancing dendrites at certain values of film thickness. They called this value the critical film thickness, and argued that it would mark the transition from spongy shedding to spongy nonshedding growth.
 The model predictions of sponginess agree reasonably well with two data sets as described in section 8 above. In addition to these favorable comparisons with the experimental data, the predictions of the model are in general agreement with other findings that suggest that the liquid fraction is rather insensitive to growth conditions and is typically in the range of 0.3–0.5 [Gates et al., 1986; Lock and Foster, 1990].
 The model-predicted sponginess shows no clear systematic error in comparison with the data plotted in Figures 5 and 7 (J. Shi, unpublished data, 1996). In contrast, the model-predicted icing flux is systematically larger than the experimental icing flux in Figures 6 and 8. This systematic error appears to be more pronounced for experimental icing fluxes below about 0.025 kg m−2 s−1. A possible explanation for this overprediction is that the model heat loss from the icing surface is too large. For example, if the impinging spray temperature assumed in the model were lower than the actual spray temperature, the model-predicted sensible heat loss from the icing surface would be too large. This in turn, would give a large value for the growth rate of pure ice and with sponginess taken into account, the predicted icing flux would be even larger than the actual icing flux. Thus experimental data that lacks impinging spray temperature measurements may confound the comparison with the model's prediction of icing flux.
 The comparison with the fitted ice fraction curve of Lesins et al.  shown in Figure 9 is qualitatively good, but the details are problematical. The model's glaze ice predictions (f = 1.0) at low liquid water contents around 2 g m−3 or less agree reasonably well with the Lesins et al.  fitted curve, where f = 1.0 at approximately the same liquid water content. The model underpredicts at low liquid water contents, and overpredicts at liquid water contents above about 8 g m−3. The reason for this discrepancy remains elusive. However, part of the difference, at least in the spongy, nonshedding regime, may have to do with loss of liquid through splashing in the experiments. This is reflected in measured net collection efficiencies as low as 0.2 at high liquid water contents, and warm air temperatures [Lesins et al., 1980]. Having no way to address the splashing problem in the model, we assume that the net collection efficiency is unity. It would be reasonable therefore to compare the model and experimental results under the same net impinging liquid flux. Hence, our model results should really be compared with the Lesins et al. results at higher liquid water contents, to allow for loss of water due to splashing in the experiments. This should improve the agreement in the spongy, nonshedding regime.
 If the ambient air temperature decreases sufficiently, the potential accretion mass flux of spongy ice will increase to the point where it equals or exceeds the available surface flux of liquid. In this regime, the model predicts a spongy accretion without liquid shedding. Blackmore and Lozowski  postulate that this will occur at a certain critical liquid film thickness in spray icing. The present model demonstrates this critical film thickness, and models the transition from “wet” spongy growth (with an organized flowing surface film) to nonshedding spongy growth (with essentially no surface liquid). If the air temperature is decreased still further, the model's sponginess decreases until “dry” ice growth occurs with no surface liquid or entrapped liquid.
 If the air temperature is increased to values near the freezing point, the model's predicted sponginess also decreases. This occurs as a result of reduced environmental heat loss and supercooling in the surficial film. If conditions include enough excess surficial liquid and an air temperature near enough to the freezing point, glaze icing is predicted, with its characteristic solid accretion and no liquid entrapment.
Kachurin and Morachevskii  derive a mathematical model of spray ice accretion for a horizontal planar icing surface. They assume growth of solid ice under a wind sheared water film that is turbulent due to the large airspeeds characteristic of aircraft icing. They present a mathematical analysis that suggests that the surface liquid water film becomes unstable at a particular critical film thickness. They also suggest that if the thickness of the surface film becomes less than this critical film thickness, then dendritic growth incorporates the entire flux of collected liquid. In their model, a transition occurs from solid ice growth under a turbulent water film to an icing condition in which “the droplets impinging on the solid body crystallize individually without merging into a continuous stable film.” Kachurin and Morachevskii  also describe this transition as occurring from a hard solid accretion to an opaque heterogeneous accretion and, as such, seem to describe a transition from glaze to rime icing. Our model, on the other hand, typically predicts an intermediate regime consisting of a laminar film and spongy growth.
 If we were to input into our model impinging droplet temperatures equal to the air temperature as Kachurin and Morachevskii  do, our predicted accretion would be even more spongy rather than solid. In addition, our model suggests that higher airspeeds would generate increasing sponginess. In this way, our model suggests that aircraft icing conditions may potentially generate very spongy growth with a larger accretion than if the accretion were glaze, as Kachurin and Morachevskii  suggest. The difference in icing regimes between the two models may have something to do with the absence of the effect of wind stress on our model's falling film and the absence of high dynamic pressures that could play a significant role in driving an internal liquid flow within the spongy matrix. Specific tests should be carried out to confirm this possibility, and the effect of wind stress and dynamic pressure should be included in our model before aircraft icing conditions are examined.
 A new theoretical model of spray icing on a vertical stationary cylinder has been developed. It is based on the morphological assumption of two-dimensional dendrites with semicircular cross-sectional tips, growing ahead of a 0°C isothermal icing interface. The dendrites penetrate a supercooled water film, grow in a stable fashion, and entrap surface water to produce spongy ice. The model also predicts a spongy icing transition from the spongy shedding to the spongy nonshedding regime. The model predicts spongy shedding ice growth, spongy nonshedding growth without surface liquid, as well as glaze icing, dry icing and no icing.
 The model's prediction of ice fraction compares well with the data of Lesins et al. . A comparison of model predictions of sponginess and icing flux with experimental data from the University of Alberta (J. Shi, unpublished data, 1996) produced good results, as well. These results are particularly good, considering the uncertainty in ice fraction or sponginess measurements of ±0.1. Nearly all the model predictions lie within this range of uncertainty of the data. Nevertheless, there are significant physical differences between the model and the experiments. Consequently, we offer a caveat concerning the indiscriminate use of the model to predict icing sponginess.
 An important source of the discrepancy between model and experimental results is the assumption used to estimate the impinging spray temperature in the model. No experimental investigation of spongy spray icing has yet included the measurement of this parameter, and we recommend that this be done for future spongy icing investigations. Also, no experimental investigation has directly observed the morphology of ensembles of dendrites at the surface of a spongy accretion. We recommend this for future modeling of the dendrites in spongy growth.
 The aircraft icing model of Kachurin and Morachevskii  includes the effect of wind stress at the surface of a horizontal flowing water film. They predict a transition from glaze icing to rime icing at airspeeds typical of aircraft icing. By extending the model of Lozowski et al.  to include the theoretical dendritic growth model presented here along with a model of wind stress, the accretion regimes typical of aircraft icing could be investigated. Such a model would be more directly comparable with available cylinder icing data.
 Since spongy accretion is well known to occur under conditions of sea spraying, an extension of the present spongy growth model to include saline spray would also be valuable. Because of a possible similarity in dendritic growth between freshwater spray icing and sea spray icing, it may not be too difficult to extend the present model to include the presence of brine at the icing surface.
 We thank John Shi and Victor Chung of the University of Alberta for sharing with us the results of their unpublished Marine Icing Wind Tunnel experiments. We also thank C. A. Knight for his valuable comments on the manuscript. This work was funded in part by the Academy of Finland, Vaisala Inc., and the Natural Sciences and Engineering Research Council of Canada.