Computation of experiments

Numerical simulation of run 573 For the computation of run 573, an effective collection efficiency γeff of 0.126, which is taken from the experiments, is applied. Since the wet-bulb and the static temperature are above Tm, a substrate heat flux is used which values 42 W/m². The composition of the ice/water layer at the substrate determines the connection strength of the layer with the substrate. A freezing initial liquid layer leads to a mechanical interlocking between the ice and the substrate’s surface and hence to a strong connection. Contrary to that, liquid water in that region is not capable of transmitting forces and if a closed liquid film emerges, the ice layer is said to shed of the substrate. In order to quantify the conditions at the substrate, a shedding criterion KT

is introduced, which is the ratio of melted liquid water plus water of the film which has not been displaced by ice particles to the available volume in the initial liquid film. This shedding criterion is computed parallel to the calculation of the layer growth and becomes one if a closed liquid layer is present at the substrate. As a result, the ice accretion is removed in the simulation if KT reaches unity.

Accretion and shedding of a porous ice/water accumulation: Detailed ...

Figure 4.3 shows the temporal evolution of the ice layer height H, the surface temperature of the substrate ϑsub and the shedding criterion KT during the first 20 s of the run. In this figure dashed lines mark the shedding of the ice layer which is the end of a growth cycle. The first six growth cycles are very short due to the initially high substrate temperature of ϑsub = ϑwb = 2 ^◦C which leads to a high heat flux from the substrate which rapidly melts the ice close to its surface. As a result, the mechanical interlocking between ice and substrate is lost.

During the first growth cycles no significant icing is observed and the substrate is cooled down almost to 0^◦C. In the beginning of each cycle the temperature rises for a short period of time which is due to convective heat inputs in the absence of a melting ice accretion. Subsequent impingement of ice crystals cools the surface again. If the mass flux of ice is high enough to compensate for the convective heat input, the substrate is gradually cooled down eventually to the melting temperature allowing an ice accretion. When the substrate is cooled down and only temperature gradients due to the applied substrate heat flux ˙qsub remain, the growth becomes (quasi-)steady-state resulting in cycles of uniform length.

The lower graph shows the shedding criterion KT and thereby the evolution of a water film at the substrate over time. At the beginning of each cycle, water is frozen or replaced by impacting cold particles which leads to an icing of the substrate’s surface. Subsequently heat fluxes due to convection and a heat flux from the substrate start melting the ice in its proximity resulting in an increase of the shedding criterion KT.

The layer thickness of the entire run is shown in Fig. 4.4 and compared with the experimental data. Numerically obtained growth and shedding is in very good agreement with the experiments. The first six growth cycles are the ones shown in Fig. 4.3 followed by growth cycles 7-10 with significant accretion.

Figure 4.5 shows the typical development of the shedding criterion for a growth cycle with observable thickness, i.e. cycles 7-10. Initially the melting rate is constant yielding a linear increase of KT. At some point the slope of the shedding criterion and hence the melting rate at the substrate decreases significantly. This transition of the melting behavior is a result of the insulating effect of the ice layer. Several heat fluxes act on the area close to the substrate in the beginning: the convective heat flux ˙qC,ev, which combines the effects of convective heat and mass transfer, the heat flux due to impinging particles and droplets ˙qW and the heat flux from the

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4.3 Results and discussion

0.5 0.75 1

0 5 10 15 20

KT(-)

t (s) 0

1 2

ϑsub(◦ C) 0 1 2

H(mm)

Fig. 4.5

Figure 4.3: Initial stage of the ice accretion. High frequent shedding and cooling of the substrate. Dashed vertical lines denote shedding events. Conditions are for run 573 from table 4.1.

substrate ˙qsub. As the thickness of the ice layer increases due to impinging ice crystals and water droplets, the area near the substrate becomes more and more shielded from the heat fluxes which affect the upper surface of the accretion. Therefore, the influence of ˙qC,ev and ˙qW diminishes.

At this point only a heat flux coming from the substrate is capable of melting the ice layer connecting the substrate with the accretion resulting in a decreased melting rate. The critical layer height at which no heat is transferred anymore is of the order of magnitude of a few particle radii and depends on the particle shape and the layer’s packing density as well as its composition. This transition in the melting behavior is typical for thick ice layers.

Accretion and shedding of a porous ice/water accumulation: Detailed ...

0 5 10 15 20

0 50 100 150

H(mm)

t (s)

1-6 7

8 9

10

Figure 4.4: Accretion thickness of run 573 over time. Solid lines correspond to numerical results while dashed lines represent experimental data.

Numerical simulation of run 543 The layer height of the well-adhered ice accretion observed in run 543 is depicted in Fig. 4.6, numerical results were obtained assuming an effective collection efficiency of 0.075 and are shown as a solid line. No heat flux in the substrate is applied since this run’s wet-bulb temperature lies below Tm. Therefore, shedding occurred only during the cooling of the substrate from initially +2 to 0 ^◦C as shown in Fig. 4.7. Since no heat flux in the substrate is applied as a boundary condition, the shedding criterion tends towards a constant value after the ice layer height passes the critical value of thermal insulation of the substrate’s surface. This means that the phase composition at the surface of the substrate remains constant. The result is a non-shedding ice accretion as observed in the experiments.