Cleaning of thick viscoplastic soil layers by impinging water jets

8 The removal of thick (approximately 1-10 mm) layers of three viscoplastic soft-solid food-9 related materials (a moisturising cream, Biscoff spread and smooth peanut butter) from 10 polymethylmethacrylate plates by turbulent, coherent, 2 mm diameter water jets was studied 11 for jet velocities of 10.6-25.4 m s -1 (10 500 < Re < 25 400). When the layer thickness was 12 smaller than the nozzle diameter, soil removal involved the growth of a circular crater. With 13 thicker layers, the removed soil initially formed a blister which subsequently ruptured, and 14 removal by cratering followed. Blister formation and dynamics were studied for the 15 moisturising cream using 2 mm and 3 mm nozzles, which indicated that this process could be 16 described by simple geometric models. Tests with the moisturising cream on glass and stainless 17 steel substrates indicated little effect of substrate on the removal behaviour. The rate of removal 18 in the cratering regime was quantified by the kinetic model of Glover et al . (2016) and the effect 19 of layer thickness on the kinetic parameters compared with the trend predicted by the model of 20 Fernandes and Wilson (2020): qualitative agreement was evident. The results were used to 21 revisit the study of blister formation by water jets impinging on Carbopol ® layers reported by 22 Tuck et al. (2019, 2020) and extract kinetic parameters from their data.


INTRODUCTION
Impinging liquid jets are widely used in cleaning-in-place operations in the food and bioprocessing sectors to clean the internal surfaces of tanks and other process vessels.The jets can be created by spray balls, static or rotating nozzles, or spinning heads.The flow of liquid serves to (i) wet the residual product, soil layer or fouling deposit on the walls, distributing cleaning agents (Morison and Thorpe, 2002) and promoting reactions or changes which facilitate removal (e.g.soaking, see Yang et al., 2019), and (ii) generate hydraulic forces which drive removal (Joppa et al., 2019).The hydraulic forces can be associated with impact, e.g.
where a droplet or jet strikes the wall (Rodgers et al., 2019), or with floweither as a fast moving film flowing radially outwards from the point of impingement or as a falling film (Wilson et al., 2012).The forces exerted on the soil layer by falling films are small by comparison with those generated in the radial flow region and higher rates of removal are therefore observed in the latter region.Rotating nozzles exploit this behaviour by moving the point of impingement over the surface to be cleaned (Bhagat et al., 2017).
When a liquid jet impinges normally on a static soiled surface the radial flow clears the soil away in an approximately circular pattern.Fingering and other phenomena can occur which give rise to complex removal patterns (e.g.Hsu et al., 2011Hsu et al., , 2014;;Walker et al., 2012).The removal of thin food and fast moving consumer goods (FMCG) soil layers by impinging turbulent liquid jets has been studied and quantified for normally impinging jets, e.g.Wilson et al. (2014), and for obliquely impinging ones, e.g.Wang et al. (2015), Bhagat et al. (2017).
These quantitative studies have shown that for a normally impinging jet the local rate of removal based on the radius of the (circular) cleared region, a, could be described well by a kinetic expression of the form (Glover et al., 2015) where t is time, k' is a rate constant, M is the rate of flow of momentum per unit length in the liquid film, and My is the value of M associated with yield stress behaviour if the fluid is viscoplastic: the rate approaches zero as  →  y .
Fernandes and Wilson (2020) identified three geometric cases for impingement, linking the local thickness of the liquid film, h, to that of the soil layer, o, as shown in Figure 1.They presented a theoretical basis for Equation (1) for the thin layer regimes, where  o ≲ ℎ, based on viscous dissipation in the soil at the cleaning front.For turbulent liquid jets h is normally smaller than the jet diameter.Very thin viscoplastic soil layers were investigated using a sheardriven approach by Fernandes et al. (2022), following the approach for Newtonian layers developed by Yeckel and Middleman (1987).

Figure 2 here
In this paper the formation of blisters in thick layers of viscoplastic soil layers and the subsequent growth of cleared regions is studied systematically using normally impinging jets, to establish the effect of layer thickness on layer behaviour and the rate of removal.Removal is monitored from above and below the layer, using transparent targets, which allows some insight into the cavern shape and behaviour.Three different soil materials are studied, and the Tuck et al. results are discussed in the light of the findings.
Detailed modelling of jet penetration and blister formation is not presented.Semenov and Wu (2016) reported a modelling framework to describe the jet penetration studies of Uth and Deshpande (2013): to describe the behaviour reported here their moving boundary formulation would have to be extended to include the presence of the rigid wall and the deformation and rupture of the blister cap.

Jet cleaning apparatus
Soil layers of uniform thickness were prepared on transparent polymethylmethacrylate (PMMA) target plates and subjected to the impact of a coherent, turbulent water jet.For all the tests reported here, the jet impinged on the soil normally.The apparatus was based on that reported by Chee and Wilson (2021) Figure 3 shows a photograph of the arrangement.Cameras mounted above and below the target plate allowed the size of the region deformed by the jet to be recorded over time.Graticule tape affixed to the target plate allowed the length scale to be determined directly from images.
Individual frames were captured at 60 fps and subsequently analysed using MATLAB TM codes to calculate the mean radius observed from above, labelled b (captured by a GoPro HERO7 Black camera), and below, labelled a (captured by Nikon D3300 DSLR via a 45° mirror).The terminology is chosen to be consistent with previous work.Figure 4 shows an example of the analysis of a pair of images: the mean radius was calculated from at least 8 measurements made at regular intervals.

Figure 3 here
Figure 4 here

Soil layers
Soil layers with thicknesses up to 9.3 mm were prepared by spreading the materials in 110×110 mm cavities in recessed plates (recess depths of 1, 1.7, 3.1, 5.3, 7.7 and 9.3 mm).After loading the surface was levelled using a scraper blade, and the mass of material measured to check reproducibility.The test materials included two hand creams (Nivea Crème ® and Nivea Soft ® ), smooth peanut butter (Sainsbury's supermarket) and Biscoff spread.The materials' rheology was determined separately using rotational rheometry (employing serrated parallel plates), using the methodology reported by Fernandes et al. (2021).A summary of the rheological studies is provided as Supplementary Material.All exhibited quasi-viscoplastic behaviour, with significant creep at low shear stresses and a transition between creeping and flowing states that takes place over a range of shear stresses.The term 'critical stress', denoted  c , is used rather than 'yield stress' to identify a characteristic stress during the transition.
After testing, the plates were wiped clear of residual soil, cleaned thoroughly using an aqueous detergent solution, then prepared for coating by rinsing with deionised water followed by isopropyl alcohol and drying.

Cratering vs. Blistering
Three modes of initial removal behaviour were observed, illustrated in Figure 5  the jet tunnelled directly though the layer until it reached the plate and spread outwards thereafter, removing all the soil and creating a crater which increased in size over time (Figure 5(a)).This is phenomenologically similar to the behaviour reported by Uth and Deshpande (2013).The liquid left the plate as a broken sheet, initially directed upwards and later, at large a, as a film climbing the crater wall.

Figure 4 here
In blistering, the jet pierced the layer, tunnelled through to the plate and formed a cavern there, forcing the soil away from the wall and forming a blister (Figure 5(b)).Some of the liquid was observed to leave through the entry hole as time progressed.After time  b the confining soil shell ruptured, often at points of weakness at its base, generating fissures through which liquid escaped (see Supplementary Figure S.5).At higher flow rates the jet then removed the remaining soil shell and removal was similar to cratering.This is the behaviour reported by Tuck et al. With Biscoff and peanut butter layers, the shell remained intact and moved with the cleaning front, termed walling (see Figure 5(c, ii)).
At lower flow rates, pooling was observed, where the water pressure in the liquid in the impinged region did not support the formation of an inflated blister.Water penetrated under the soil layer and also flowed over the top of the soil.
Table 1 summarises the observed cleaning behaviour on Perspex surfaces.Cratering was observed for all the materials tested in this work when  o <  N , whilst some form of blistering was observed with thicker layers, indicating that the transition was associated with the geometric criterion  o  N ⁄ = 1.This is consistent with the results of Tuck et al., who reported blistering for all their tests, which featured  o  N ⁄  1. Cratering was observed in previous tests performed on thin layers of Carbopol ® gels in our laboratory (data not reported).
Table 1 here The influence of surface material on the removal of Nivea Soft was tested using 3 mm diameter jets and three soil layer thicknesses ( o  N ⁄ = 1, 2 and 3).The summary in Table 2 shows that pooling was observed with all thicknesses for all three materials at a flow rate of 1 L min -1 .
Blistering was observed for all three thicknesses at higher flow rates on glass and Perspex.The behaviour on stainless steel was similar, except that pooling was observed with the thickest layers at 2 L min -1 with this material, indicating that there is some adhesive contribution to the mechanism.

Table 2 here
The influence of soil rheology is characterised here using the ratio of jet inertia to critical stress,  =  o 2 2 c ⁄ , as discussed by Blackwell et al. (2015) and Sen et al. (2020) (Piau, 2007) and the latter being an oil-in-water emulsion (Nivea Beiersdorf, 2022).The condition for rupture of the blister is not known, and is expected to be related to the confining shell of soft solid reaching a critical size, which in turn will be related to the volume of the cavern reaching a critical value.In the absence of a detailed criterion, the ratio of the

Figure 8
The above results indicated that rupture is associated with the volume of the cavern.Figure 9 shows that the point at which Nivea Soft ® blisters ruptured, expressed in terms of the volume of liquid which had been delivered, depended strongly on the initial layer thickness and, to a lesser extent, on the jet flow rate.The nature of the surface did not have a systematic effect.
Pooling was widely observed at the lowest flow rate tested, 1 L min -1 , and the associated liquid volumes differed noticeably from the trend at larger flow rates.Biscoff and peanut butter blisters did not rupture in the same way: in walling, the shell remained intact and expanded with the cleaning front.

Figure 9
The above results can be used to relate the volume of liquid supplied to the blister at the point where it bursts, Qt, to the initial thickness of the layer,  o .Assuming that  = 45° (so that  =  −  o ) and solving Equations ( 2) and ( 3 from a simple wedge: for all the cases they reported, and for other viscoplastic soils considered by Fernandes and Wilson (2020), the angle at base of the crater was less than 45°.For thinner soil layers, the contribution from the base of the rim to the total volume of the displaced soil is expected to be more significant, and this would explain the discrepancy between the liquid volume added at bursting predicted by Eq. ( 3) and the experimental values evident for thinner soil layers in Figure 9 Tuck et al. (their Figure 7) reported tb values of 0 s and 1.5 s for Carbopol ® layers with  = 2 mm and 8 mm, respectively for a water jet with Q = 470 mL min -1 and dN = 1 mm.These results are consistent with the findings in Figure 9: the volume at rupture for  = 8 mm, approximately 5 mL, corresponds to a time of 0.8 s.There would be a delay before the roof of the cavity was cleared and the black base was visible, which was their criterion for rupture.

Cleaning rates -Stage I
The geometric model indicates that removal of the soil layer in Stage I is driven by the growth of the liquid void.The values of the estimated ratio of stagnation pressure to yield stress in Table 1 are all large, indicating that the resistance of the soil layer to deformation is not likely to control the initial removal dynamics.At longer times, the expansion of the blister shell and widening of the entry hole allowed liquid to escape so the volume of liquid contained is not known, and will be accompanied by a reduction in the pressure to expand the cavity.
The rate of soil removal was calculated from the change in the size of the cleared area, da/dt.
Figure 10 shows the dependence of this rate on a for one case: similar behaviour was observed with Nivea Soft ® , peanut butter and Biscoff where blistering was following by walling or cratering.There is a noticeable change in behaviour as the cleaning front reaches a = 10 mm, corresponding to the transition from Stages I to III (Figure 6).(4) Figure 10 shows that the initial removal rate is estimated reasonably well by this result.As a increases, some of the water leaves the blister so Equation ( 4) is expected to overestimate the rate.This is evident in the Figure for a > 5 mm.The fraction of liquid retained would then be determined by the shape of the exit hole at the top of the cavern: modelling this process was not attempted here.Similarly, detailed fitting of angles  and  was not attempted.

Stage IIIcrater growth
One reason for not conducting more detailed modelling of Stage I is that in practice the point where the jet impinges the surface moves, as discussed by Glover et al. (2016), so that once a blister ruptures, removal is determined by the mechanism active during the cratering or walling phase.At times when removal involved cratering or walling, Equation (1) was found to fit the data reasonably well, subject to increasing scatter with thicker layers.Figure 11 shows examples of the fit to the model for the different layer thicknesses for peanut butter, and the scatter in the data is very noticeable for 0 = 7.7 and 9.3 mm.The data sets cover different ranges of M as the onset of stage III and the value of My differs in each case.

Figure 11 here
In these calculations, the local value of M was estimated using the Wilson et al. (2012) model and the values obtained for k′ and My for Nivea Soft ® , Biscoff and peanut butter layers with dN = 2mm and 2 L min -1 are reported in Figure 12.The general trend is that k′ decreases with layer thickness and My increases, both of which are associated with thicker layers being cleared more slowly.The absolute values are comparable, which is consistent with the rheological parameters being of similar magnitude.The values of ′ decrease with increasing critical stress of the soils, which confirms that the rheology of the soil plays an important role during cleaning (as reported by Fernandes et al., 2019).
Equation ( 1) was developed for thin layers and its extension to cratering for thick layers has not been investigated in detail to date.Fernandes et al. (2021) presented an extension of the Fernandes and Wilson (2020) viscous dissipation model to include thick layers of a Bingham fluid, obtaining the relationship (with a number of simplifications) where B is the shear rate dependency of the Bingham fluid and i is the thickness of a residual layer.In this model the cleaning front is assumed to take the form of a straight ramp inclined at angle  to the horizontal (see Figure 1(b)).Note that in this paper, angle  is a similar quantity but is estimated from experimental measurements (see Figure 5(b)).This predicts a decreasing trend with increasing o which does not describe the data in Figure 12(a) that well at larger values of  o , suggesting that other effects not captured by Equation ( 5) are involved.The Fernandes and Wilson (2020) model assumes that the thickness of the liquid film (and hence the calculation of M) is not affected by the presence of the berm of soil downstream, which is unlikely to hold when  o  N ⁄ is large.

Figure 12 here
The values of My show a systematic increase with soil thickness which is consistent with the dependency in the Glover at al. (2016) and Fernandes and Wilson (2020) models, viz.
This result is based on a static force balance when the presence of the yield stress has stopped the growth of the cleared region.There is noticeable scatter in the data so the dependency on  c is not clear.One of the sources of the scatter is that the parameters ′ and My are correlated, being extracted from plots of da/dt versus M, so variation in one affects the estimate of the other.Equation ( 6 Figure 13 shows the effect of jet flow rate on k′ and My for Nivea Soft ® layers.The k′ values for Q = 1 L min -1 in Figure 10(a) do not follow the general trend exhibited by the other flow rates: pooling behaviour was observed at this flow rate and the hydrodynamics are expected to differ from the model.For the other flow rates, the similarity in values indicates that the effect of flow rate on the cleaning rate is captured in the calculation of M. The finding that Equation (1) provides good quantitative agreement with the observed removal behaviour indicates that it can be used in a semi-empirical fashion to predict cleaning rates, e.g. for jets generated by moving nozzles. Figure 13(b) shows the values obtained for cleaning by jets generated by 3 mm diameter nozzles.Similar trends are evident, with reasonable agreement for different flow rates, but the absolute values differ by a factor of 2 or more.This difference is attributed to the use of the Wilson et al. (2012) model to calculate M in this work: this model does not include the development of the momentum boundary layer in the thin film (it gives an analytical result not requiring detailed numerical evaluation).Analysis of the data sets using more detailed models such as that of Bhagat and Wilson (2016) should be performed to determine the relationship between flow rate, jet diameter and soil layer thickness.

CONCLUSIONS
The removal of thick layers of soft solid material from PMMA plates by turbulent impinging cylindrical water jets was studied systematically using a two-camera system.This arrangement allowed some of the observations reported by Tuck et al. (2019Tuck et al. ( , 2020) ) to be explained.
Blistering was observed in the initial stage when (i) the layer thickness was greater than the jet diameter, and (ii) when the ratio of the intertial stress to the critical flow stress, IF, was large.
Pooling was observed when IF was not large.These results indicate that for large IF the phenomenon is driven by geometrical factors.Analysis of the blistering dynamics showed that the behaviour in this stage was linked to the volume of water contained within the blister rather than hydrodynamics, with rupture occurring when a critical volume (or a critical level of strain in the confining shell) is reached.
The removal rate data in the cratering stage could be described reasonably well by the kinetic model presented by Wilson et al. (2014), for all three soft solid materials employed in these experiments.The observed dependency of the model parameters on flow rate and layer thickness followed the expected trend but quantitative agreement of the parameters with the detailed model of Fernandes et al. (2021) developed for thin layers -was no better than fair.
Further work on detailed models is required.
The results provide mechanistic insight into the experiments on Carbopol TM layers reported by Tuck et al. (2019Tuck et al. ( , 2020)).The removal data for the cratering stage in their tests were analysed using the Wilson et al. (2014) model and gave poorer agreement with the model than the materials tested here, which is not surprising given the systematic uncertainties involved in the analysis.).Data were fitted to effective radius (/) 1/2 for times after the evident kink in the profiles (point d in Figure 2).Error bars are not reported owing to the systematic uncertainty in the approach.

Figure 1 here
Figure 1 here

Figure 6
Figure6shows an example of blister dimensions for a case where a blister formed initially and

Figure 6 here
Figure 6 here

Figure 7 here
Figure 7 here volume of soil layer dislodged by the jet to the volume of water added (assuming no water leaves the cavern), VR, was calculated.The former was estimated as the volume of a frustrum of radii a and b, height o, Figure 5(b), viz.for Nivea Soft ® with dN = 2 mm are plotted against b (which increases with time) in Figure 8.The data for Q = 1 L min -1 are not plotted as these were associated with pooling.Similar plots were obtained with dN = 3 mm.VR decreases steadily with time and the trajectories terminate when VR ~ 1.0-1.2.
) simultaneously for a given value of VR yields the cubic relationship between Qt and  o plotted on Figure 9.The agreement at lower  o is poor and this represents a feature that requires further investigation.The morphology of the cavern may differ from the frustum assumed by Eq. (3), particularly for thinner soil layers.Fernandes et al. (2019) demonstrated that the rims for the thin soil layer cases (see Figure 1(b)) differed

Figure 10 here
Figure 10 here ) suggests that the data for different materials should follow the relationship  y ∝  c  o , which the inset in Figure 12(b) shows holds for these data.These results indicate that effort to modify the Fernandes and Wilson model should focus on the derivation of the ′ term.
Figure13 here Figure 14 here

Figure 2 Figure 3 Figure 4 Figure 6
Figure 2 Example of growth of cleared region reported by Tuck et al. (2020) for a 2 mm thick Carbopol ® layer on a black painted, stainless steel plate subjected to a water jet (Q = 600 mL min -1 , dN = 1 mm, room temperature, Re ~ 13 000) as indicated in inset

Figure 7 .Figure 8 Figure 9
Figure 7. Evolution of features during blistering of Nivea Soft ® , Stage I, dN = 2 mm.(a) blister radius b; loci show predicted evolution of b for a spherical cap (see inset) assuming no liquid loss and constant , Equation (2); (b) internal angle  (see Figure 5(b)), legend indicates Q and o values.Inset shows evolution of estimated yielding wedge shape for a 9.3 mm thick layer, Q = 2 L min -1 .

Figure 10 .
Figure 10.Evolution of cleaning rate with radial position, for repeat runs under conditions in Figure 6 (9.3 mm thick Nivea Soft ® layer, dN = 2 mm, Q = 2.4 L min -1 ).Vertical dashed

Figure 11 Figure 12
Figure 11 Examples of fit, shown by dashed lines, of Equation (1) to rate of cleaning (as viewed from below, da/dt) in Stage III for peanut butter layers with initial thickness indicated.Note that M decreases with time, as a increases.Q = 2 L min -1 , dN = 2 mm.

Figure 13
Figure 13 Effect of jet flow rate on Nivea Soft ® cleaning rate parameters with (a) dN = 2 mm,

Figure 14
Figure 14 Effect of layer thickness on cleaning model parameters (a) ′ and (b) My extracted and summarised in Table 1.Examples of videos of each case are available from the University of Cambridge Apollo repository at[weblink].Cratering was observed when  o <  N for the three softer soils: in their studies of viscoplastic drops impacting on coated surfaces.The Carbopol ® gels used by Tuck et al. had a noticeably smaller critical stress, of 3.42 Pa, than the other materials in Table 1, giving IF values of order 1000: the Carbopol ® layers can be classified as soft, as the resistance of the material is small compared to the inertia of the jet.The IF values for Nivea Soft ® tests ranged from 12 to 340, thereby spanning the values of the other materials.Inspection of Table 1 indicates no effect of IF when  o  N ⁄ < 1.When  o N ⁄ > 1, pooling was observed on Perspex when IF < 20.A small number of tests were conducted with Nivea Crème ® , with IF ~ 15, and only cratering was observed with this material.Further tests were not conducted with this material as the focus of this study was on blistering.Blistering or walling, i.e. interrupted blistering, was observed consistently at larger values of  o  N ⁄ and IF.For intermediate values, the observed behaviour could not be classified simply on the basis of  o  N ⁄ and IF: these initial findings indicate that IF is a useful quantity for estimating layer behaviour, reflecting the findings of Blackwell et al. and Sen et al. for drop impact.Nivea Soft ® soils were investigated further as these mimicked the behaviour observed by Tuck et al.Both the Carbopol ® dispersion investigated by Tuck et al. and Nivea Soft ® are composed of microstructurally arrested (jammed) systems of soft particles, the former being swollen microgel particles