Basic theoretical concepts

The basic problem, i.e. the rate of icing (mass growth per unit time and unit area), can be formally expressed simply as

where W is the liquid water content in the air and V is the wind speed. WV gives the mass flow of supercooled water in the air. The coefficient e is the collection efficiency and is a function of properties of flow and icing structure. The term (1−n) expresses the portion of impinging water that freezes. ’Dry growth’ (n= 0) is when all the water freezes, and ’wet growth’ (0 < n ≤1) when all the latent heat cannot be transferred from the freezing surface. Hence, the term n may be interpreted as a resistance expressing the part which does not freeze and runs off. The accretion efficiency (1 −n) is a function of the mechanisms controlling primarily the heat balance at the freezing surface. The most important meteorological quantities involved are air temperature T _a, wind speed V, and liquid water content W. Accordingly, the rate of ice accretion may be expressed by the following quantities

where, in addition to those given above, the quantities F and S characterize the properties of flow and structure, such as the effect of free-stream turbulence and form, size, and material properties of the icing structure.

The heat balance of a freezing surface, controlling the rate of icing, has been discussed in several papers (Reference LudlamLudlam, 1951; Reference MessingerMessinger, 1953; Reference Cansdale and McNaughtanCansdale and McNaughtan, 1977; Reference Lozowski, Lozowski, Stallabrass and HeartyLozowski and others, 1979, Reference Lozowski, Lozowski, Stallabrass and Hearty1983[a]; Reference MakkonenMakkonen, 1981). Considering the most important heat-exchange mechanisms, the quasi-stationary heat-balance equation for a freezing surface can be given as

and

where R is the rate of ice-mass growth, L _f is the latent heat of freezing, and the other fluxes increasing or decreasing the enthalpy of the freezing surface are defined (cf. Notation)

The physical basis and expression of the different processes above correspond closely to those given by Reference Cansdale and McNaughtanCansdale and McNaughtan (1977) and Lozowski and others (1979, 1983[a]). Therefore, only a few observations seem to be relevant here. First, the radiative heat flux was taken into account using the net radiation method (cf. Reference IsachenkoIsachenko and others 1977), assuming both emissivities ϵ_s and ϵ_a to be around unity. This assumption may yield somewhat underestimated radiative-flux results, because of a limited optical length in the experimental studies, but the importance of the error in this less significant term (q1) is negligible (cf. Table I). A linearized approximation describes the term well. Secondly, in the term q _r it has been assumed that all the impinging water which does not freeze will be heated to the temperature of the surface before shedding and run-off. This may, in cases of bouncing, lead to an overestimation, which is not serious with regard to the objective of this study. Finally, in the term for kinetic heating (q _k) due to water droplets, it should be noticed that in many cases it is irrelevant to assume that all the kinetic energy converts to heat, giving rise to a proportionality factor c (<1). In fact, this may be of importance only at very high wind speeds when the kinetic energy gain from the water droplets is of practical importance.

With reference to the importance of different heat-transferring mechanisms, Table I gives the magnitude of the different terms in the heat-balance Equation (3) for typical icing conditions. The examples are given as averages for the front half of a cylinder 5 cm in diameter and using the smooth cylinder formula of Equation (8) for the convective heat-transfer coefficient h (cf. paragraph on “amount of ice growth” below).

Experiments

A description of the wind-tunnel installation and experimental procedure has been given in Part I of this paper. In these quantitative tests, PVC-coated Styrox cylinders of low thermal conductivity and heat capacity, 20 cm long and 5 cm in diameter, were used. Wind velocities varied from 7.5 to 20 m/s, thus the Reynolds number was 2.6 × 10⁴ to 7 × 10⁴. The liquid water content varied from 1 to 11 g/m³, yielding wet-growth conditions in most cases. The ambient air temperature varied between 0° and −13° C. The geometric quantities were measured from the ice deposits, and the amount of ice was measured by weighing a cut piece about 15 cm long, and homogeneous in the vertical plane.

Amount of ice growth

Because of the complicated and varying form and structure of deposits under different icing conditions (roughness, hollowness, and density variations), as revealed in Part I of this study, expression of quantitative ice growth in simple geometric quantities has considerable inaccuracies. Therefore, we used the total mass accreted (averaged per cylinder unit length or per surface unit area, taking the areal increase into account) as a primary verification basis for estimated and experimental ice growth rather than, for example, a stagnation line thickness (cf. Reference Lozowski, Lozowski, Stallabrass and HeartyLozowski and others, 1979, Reference Lozowski, Lozowski, Stallabrass and Hearty1983[b]) or a cross-sectional area of a deposit (Reference Bain and GayetBain and Gayet, 1982). Accordingly, when comparing the experimental and estimated ice growth, for the estimated ice growth Equations (3) and (4) were applied to the whole deposits, i.e. the averaged convective heat-transfer coefficient (h), averaged recovery factor (r _c), and averaged collection efficiency (e) were used (cf. section on “experimental results of the total convective heat-transfer coefficient” below). Furthermore, an averaged mass flux of water droplets R _w for a cylinder yields R _w × (2/π).

Figure 1a and b shows the rate of ice growth, in various meteorological conditions, plotted against the liquid water content. The results are derived from 15 min tests and the rate of ice growth is given as mass growth (m ₁) per cylinder unit length. The calculated product eWVb gives the total amount of water falling on the surface. This was calculated taking into account an increase of the sample width b during a test (cf. Part I of this study). Figure 1c gives a comparison between the experimental and theoretically calculated growth rates.

Fig. 1.

a. Dependence of ice-mass growth (m1) per unit length for a 5 cm diameter cylinder on the liquid water content, for various temperatures and a wind speed of 15 m/s. and test duration of 15 min. The product eVWb gives the total amount of water falling on the surface. The vertical pointers below show the theoretical critical water content (Wc) for the temperature in question.

b. Dependence of ice-mass growth per unit length for a 5 cm diameter cylinder on the liquid water content, for a set of experiments of V = 10 m/s and T = —6°C, and one of 20 m/s and −9°C. Test duration 15 min. The product eVWb gives the total amount of water falling on the surface, and the pointers below show the theoretical critical water content.

c. Comparison of the experimental and theoretically calculated ice growth. The experimental curve is based on a set (Ta= −8° C) of measurements from Figure 1a. The graphs ‘1’ and ‘2’ give the theoretically estimated ice growth, using for the convective heat-transfer coefficient the smooth-surface formula of Equation (8) in ‘1’ and the rough-surface formula by Reference Lozowski, Lozowski, Stallabrass and HeartyLozowski and others (1983[a]) in ‘2". Both theoretical graphs are given for the droplet size of MVD m 60 μm (lower lines) and 80 μm (upper lines).

In calculating the theoretical critical water contents (W _c) for Figure 1a general assumption (Reference Cansdale and McNaughtanCansdale and McNaughtan, 1977; Reference Lozowski, Lozowski, Stallabrass and HeartyLozowski and others, 1979) was adopted, according to which a surface behaves as a smooth surface during the dry-growth stage of icing of a cylinder. Accordingly, a smooth-surface formulation, Equation (8), was used for the convective heat-transfer coefficient h, and the critical liquid water content values were calculated from the maximum freezing capacity for the given conditions, i.e. as a solution of W(= W _c) through R _w from the heat-balance Equation (3) when R = R _w. The theoretical ice growth in Figure 1c has been calculated according to Equations (3) and (4), for h using the above smooth-surface formula in one case and, in the other, the rough-surface formula suggested by Reference Lozowski, Lozowski, Stallabrass and HeartyLozowski and others (1983[b]). Due to the in accuracy of the droplet-size spectrum data, theoretical curves have been calculated for MVD of 60 μm and 80 μm.

The results in Figure 1 suggest several distinct conclusions. First, the experimental critical liquid water content, i.e. a maximum W when practically all the impinging water droplets freeze, seems to be much larger than the one theoretically estimated (Fig. 1a and b). Secondly, in the wet-growth region, the rate of growth seems to depend distinctly on the liquid water content. Thirdly, the experimental graphs approach dry growth (eWVb graph) gradually (cf. Reference Carras and MacklinCarras and Macklin, 1973), which shows that a change from a dry growth into a wet growth is a transition process and not an abrupt one (cf. Reference MakkonenMakkonen, 1981). Furthermore, the visual observations have shown that, although in this region all the impinging water freezes, it is not necessary that the surface is physically dry all the time. Finally, the experiments yielded significantly higher ice growth than theoretically estimated (Fig, 1c). In order to interpret the characteristic results above, and because there is not at present any satisfactory theory, a deeper approach to the heat-exchange problems is now taken (see below).

Table I.

Theoretical Estimates Of The Different Heat Fluxes of Equation (3) For Wind Speeds of 10 and 20 m/s, Temperatures OF − 1° and − 10°C, and Liquid Water Contents of 1 and 4 g/m3

Experimental results of the total convective heat-transfer coefficient

Equation (7) offers a means for calculating from experimental results the transfer coefficient h. Because the surface temperature of a deposit could not be measured, h could only be calculated for wet growth when Ts ~ Tf ~ 0°C. The study was carried out by averaging over the deposits, thus giving the average h values. An increase in sample width during the test was taken into account in calculating the terms R, e, and R _w, using the mean of the initial cylinder width and the measured final sample width as an effective width, b, and the effective heat-transferring surface was approximated as a half cylinder of a diameter of b. This approximation fits short-duration ’tests well (cf. Part 1) but it may introduce an error in longer-period tests of over 10–15 min. In all the considerations, data obtained by a PVC-coated cylinder were used and, before applying Equation (7), the rate of ice growth R was corrected with respect to the heat capacity of the test probe in order to avoid errors in results of h in very short tests.

Figure 2 gives, in the form of a time series, the experimental h calculated from a few available sets of consecutive measurements of different durations, in stationary conditions. For reference, Figure 2 also gives h for the front half of a smooth cylinder calculated from

where k_a is the molecular heat conductivity. Re is the Reynolds number, D is the cylinder diameter, and θ the angle from the stagnation line. Equation (8) is an approximation used by Reference Lozowski, Lozowski, Stallabrass and HeartyLozowski and others (1983[a]), based on the Nusselt number data of Reference AchenbachAchenbach (1977).

Fig. 2.

Time development of the average experimental convective heat-transfer coefficient for various icing conditions. The coefficients h _s(10) and h _s(15) give the smooth-cylinder coefficients according to Equation (8).

In addition to the somewhat sparse data in Figure 2, we obtained several sets of short-period data from tests of duration 0.5−5 min under various conditions. The data from these tests confirm the results and, together with the data in Figure 2, suggest that the experimental h in wet-growth conditions is significantly greater than the theoretical one, calculated for a smooth surface, right from the very beginning of the test. When comparing common deposit size and form characteristics of our tests (cf. figures 2 and 3 in Part I) with the above, we find that during the first 1–2 min a deposit surface looks smooth but after 5 min the whole surface is covered with small roughness elements and after 9–10 min the deposit reaches its overall structure, after which it increases in size.

Fig. 3.

Fig. 3. a. Dependence of the experimental convective heat-transfer coefficient on the liquid water content in the air for various temperatures and a wind speed of 15 m/s. h_s(15) gives the coefficient for a smooth cylinder, b Dependence of the experimental heat-transfer coefficient on the liquid water content in the air for wind speeds of 7.5, 10, and 20 m/s. For symbols, see Figure 3a .

Figure 3 gives the experimental heat-transfer coefficient h versus liquid water content for various wind speeds and air temperatures, calculated from 15 min experiments. The experimental h is seen to depend distinctly on W, especially for low contents of liquid water in the air. For high wind velocities, the dependence is more apparent than for lower ones. At small liquid water contents, the graphs tend towards h _s. The method of calculating h from Equation (7) is thus "self-checking" and indicates that the overall parameterization of Equation (3) is relevant. As discussed before, h could only be calculated in our case from Equation (7) for wet-growth conditions of W ≥ W _c when T _s ~T _f. Figure 3 does not indicate any apparent dependence of h on r_a. Unfortunately, the droplet-size spectrum (or MVD) could not be measured so that the role due to its possible changes in the above results could not be investigated.

The findings given in Figures 2 and 3 raise the question: what physically underlies the facts according to which the experimental convective heat-transfer coefficient from the very beginning of icing is much larger than the theoretical one for a smooth surface, and, that this coefficient seems to be strongly dependent on the liquid water content W? In answer, we might expect that the surface roughness of an ice deposit, the effect of which on heat transfer is rather poorly known, would be principally responsible for the above. However, if we consider the problem from this point of view, we can find the following arguments which do not strongly support that assumption. First, the results show that h is large from the beginning of the icing, i.e. before the formation of the visual roughness elements, whereas during the formation of the roughness elements h increases about 20% in four cases out of five (see Fig. 2). Secondly, the dependence of h on W in Figure 3 distinctly exceeds the magnitude of time dependence in Figure 2 when the roughness elements are being generated, and Figure 4 reveals a more or less similar interdependence between h and W in short-period tests, during the early stages of the formation of roughness elements. Thirdly, according to the drag coefficient and the Nusselt number studies given in the literature, it seems that at the low wind velocities (i.e. low Reynolds numbers) considered the dependence of heat-transfer coefficient on surface roughness is rather low. Additionally, in the case of a hydrodynamically rough cylinder, a further increase in roughness at low Reynolds numbers does not significantly increase skin friction (Reference AchenbachAchenbach, 1971) and heat transfer (Reference Lozowski, Lozowski, Stallabrass and HeartyAchenbach, 1977). Actually, a classification of the roughness of an ice deposit, e.g. in terms of the so-called ‘equivalent sand-grain roughness”, is extremely difficult, as pointed out by Reference Lozowski, Lozowski, Stallabrass and HeartyLozowski and others (1983[a]). This is distinctly revealed from figures 2 and 3 in Part I of this paper. Finally, if we follow the assumptions of Reference Lozowski, Lozowski, Stallabrass and HeartyLozowski and others (1983[a]) that a measure of the highest roughness in Reference AchenbachAchenbach’s (1977) experiments represents an ice degosit in roughness, we find (Reference AchenbachAchenbach, 1977, fig. 15) that h for a rough half-cylinder is approximately the same as for a smooth one when Re≈ 4.8 × 104 and 1.6 times that of a smooth cylinder when Re ≈ 7.3 × 104. When comparing this latter result with figures 2 and 3, h obtained from this study can be seen still to be much larger but it is interesting to note that the increase of h in Figure 2 is of the same order of magnitude as the above change from smooth to rough according to Achenbach’s results. The latter Re = 7.3 × 104 corresponds, actually, to a higher velocity (of about 20 m/s) than the results in Figure 2.

Fig. 4.

Fig. 4. Convective heat-transfer coefficient versus the liquid water content for test periods of 1–5 min (V = 15 m/s). The dashed line represents the dependence of 15 min tests in Figure 3.

Summarizing all the discussion above, we conclude that the large convective heat-transfer coefficients found in this study are not primarily due to surface roughness. This does not mean, however, that surface roughness cannot have any role in this but in the following section, when trying to find a more comprehensive explanation, we just cannot explicitly separate the mechanisms from each other using the available data.

Total convective heat-transfer coefficient

In order to find an explanation to the problem outlined above, the convective heat-transfer coefficient h, which controls the fluxes of sensible and latent heat during icing, is further discussed below.

If we first consider a semi-stationary convective heat-transfer process, the temperature conditions at a heat-transferring surface, corresponding to those during icing in the dry-growth region, can be expressed schematically as in Figure 5a. In this case, the convective, i.e. sensible heat flux, is given as

where h ₁ is the transfer coefficient in question. Using a so-called (e.g. Reference KreithKreith, 1966) equivalent thickness of a conductive sub-layer, δ_a, h ₁ can be expressed as h ₁ = k_a/δ_a where ka is the molecular thermal conductivity in air.

Fig. 5.

Fig. 5. A schematic diagram showing temperature conditions: (a) in a viscous sub-layer in air above an ice—air boundary under dry-growth icing; (b) in the viscous sub-layer in water and in air in an ice-water film-air boundary under wet-growth icing.

In Figure 5b the temperature conditions are given, analogically, for the case when a water film exists above the heat-releasing surface, corresponding to the wet-growth process. In this case, the viscous sub-layer in the air is assumed to be more or less similar to the former one but, because of the water film (under which the ice grows giving rise to T _s being slightly lower than T _f), there also exists a viscous sub-layer in the water film. Because of the continuity of the transfer

the following equality must hold:

Eliminating T _s, the solution of h _t from the above gives the total heat-transfer coefficient as

where h ₂ is the transfer coefficient in the water film. As an analogy to h ₁, the coefficient h ₂ may be expressed formally as h ₂ =k _w/δw where kw is the molecular heat conductivity in water and δW is the equivalent thickness of the conductive sub-layer in the water film. Equation (11) covers both cases discussed above. In the former case, δW ⟶ 0 and h _t = h ₁, corresponding to a dry-growth process.

It must be emphasized that in the above discussion the equivalent thickness of the conductive sub-layer is a pseudo-thickness merely of practical relevance. This corresponds only to an extension of a linear temperature profile from a boundary, determined by molecular mechanisms, to the total temperature jump. Accordingly, the familiar Nusselt number for a cylinder in terms of δa is Nu = h D/ka = D/δa, by definition (cf. Reference KreithKreith, 1966).

Effect of perturbation due to water droplets impinging on a viscous sub-layer and heat transfer

Strictly speaking, the above consideration is only valid for a process due to temperature differences, and without the flow of impinging water droplets corresponding to a situation under real icing conditions. In the former case, the heat transfer is determined according to Nu =f/(Re) only. In this case, however, the flow of water droplets impinging on a surface causes perturbation in the boundary layer (cf. Fig. 5b). This can contribute to heat transfer in several ways. First, when a water droplet impinges on a surface, it penetrates the heat-transferring layer and locally destroys the structure of the viscous sub-layer in the air-ice or air–water film-ice boundary (the number of impinging water droplets was 10³–10⁵ cm − ² s − ¹ in these experiments). This perturbation, together with the effects due to splashing, enhances the heat transfer which is otherwise controlled more or less by molecular mechanisms at the boundary. Formally, this enhancement may be interpreted as a decrease in the “effective” equivalent thickness of the conductive sub-layer, similar to the known effect due to an increase in the surface roughness in convective transfer. Combined with “a droplet perturbation”, this effect, in later stages of icing, probably affects the icing process.

As a first approximation, it is reasonable to assume that the enhancement of the transfer above is proportional to the number and momentum of impinging droplets, i.e. an increase of R _w (increase of W or V) increases the convective heat-transfer coefficient in Equation (9) through an increase of h1 in the atmospheric boundary layer. An increase in the droplet size (MVD) might be expected to affect the convective transfer in a similar way. At large W flows, however, the process becomes a wet one in which the surface is covered by a water layer. Within the limits of surface structure and gravitation, the thickness of the water layer tends to thicken with increasing water flow. Although this layer is also exposed to perturbation due to the impinging droplet flow, and hence the heat transfer (h2) is controlled to a lesser degree by the molecular mechanisms, the water layer forms a limiting process for heat transfer when liquid water contents in flow are large, even though the increase in the number of droplets tends to increase h ₁ further. For example, in the stagnation region of low shear but large collection efficiency, the water film tends to thicken most rapidly thus decreasing the heat transfer, and this gives rise to the formation of a stagnation hollow. Based on these arguments, the two asymptotic regions for Equation (11) may be expressed as

for a dry process when δ_w = 0 and,

for a wet one of large W when δ_w increases and h2«h1

In these considerations we have merely tried to find an explanation for the behaviour of the convective heat-transfer coefficient. Thus we have not paid any attention to the net effect arising from a change in the droplet spectrum (increase in MVD – decrease in the number of droplets) to the form and size of an object exposed to droplet perturbation, to alignment of the surface with respect to the flow, etc. Therefore, we refrain from a parameterization of the sparse experimental data available. However, for small contents of W, the results (see Fig. 3) suggest h follows Equation (12) linearly with W, and show that h _t can be expressed as in Equation (11), as given for the two examples in Figure 6.

It is reasonable to assume that, for example, in the results by Reference Lozowski, Lozowski, Stallabrass and HeartyLozowski and others (1979, 1983[b]) this “droplet perturbation” might explain icing rates larger than the estimated ones for large W. However, the data given in the paper do not allow a quantitative approach.