Maximum Spreading of Urea Water Solution during Drop Impingement

2.1 Setup

The liquid used for the experiments is an aqueous urea solution containing 32.5 wt % urea (Merck; > 99.5 % purity) with physical properties as displayed in Tab. 1. Single droplets are produced by a droplet delivery system consisting of a syringe pump supplying liquid to a vertically arranged microcapillary stainless-steel tube. Droplet detachment is induced by gravity, while the initial droplet diameter (D₀) is regulated by taking capillaries of two different sizes featuring inner diameters of 110 and 510 µm and outer diameters of 210 and 830 µm, respectively. The resulting droplet sizes, measured by mass and verified by image analysis, are D₀ = 1.95 and 2.96 mm, respectively. Both values are close to the capillary length scale, which is 2.65 mm.

Adjustment of impact velocity (U₀) is regulated by the falling distance of the droplets independently from droplet size. Theoretical values of the impact velocities are verified by image analysis. The impingement target is mounted for perpendicular impact. The stainless-steel target features an equilibrium contact angle of θ = 50.3° for the UWS, which is measured from sessile droplets. The surface roughness of the polished steel is measured to be R_a = 0.6 µm with a Perthen perthometer M4P. A high-speed camera (LaVision HighSpeedStar 5.1, 1024 × 1024 pixel) is installed to record droplet impingement using shadowgraphy imaging technique. A laser beam captured by a photodiode is used to trigger image acquisition by detection of falling drops. Droplet spreading dynamics are monitored with frame rates of 1000 to 5000 frames per second.

Tab. 2 summarizes the parameters of six experimental cases. Each experiment is repeated three times. For these six main cases, temporal spreading curves averaged over the three realizations are presented below. For further experiments with identical droplet size but different impact velocities not listed in Tab. 2, results for the maximum spreading radius will be given only. The variation of the maximum spreading radius in the three realizations is very small with a deviation of less than 2 % only. For all cases, the droplets deposit on the solid surface without splashing.

The Reynolds number Re, Ohnesorge number Oh, and Eötvös number Eo defined in Eq. 1 serve together with the Weber number We = Re²Oh² for comparison with literature models.

(1)

2.2 Image Analysis

An image analysis script for automatic evaluation of the spreading dynamics was written in Matlab. In the first step of this routine all images showing contact between droplet and wall are distinguished and re-labeled. After contrast adjustment, a gray level threshold is calculated from the first image in a series. Based on this, all images are converted to inversed, binary images followed by a connected-components analysis and a filling operation. In case of small deviations in the horizontal arrangement of the impingement target or camera view (impact angle 90°), an inclination correction and image truncation is performed. An automatic detection function for the droplet contour delivers pixel data of the droplet spreading length, which is extracted for each image. The spreading length is converted to metric units using a scaling factor of 0.0122 mm per pixel, which was determined by calibration of a reference image. By this scaling factor, a droplet diameter of 1.95 mm is represented by 161 pixels.

3.1 Governing Equations and Code Implementation

For the numerical simulations, the phase-field method is used ^41-43 which relies on the concept of free energy, consisting of two contributions associated with the bulk and interface regions. The phase distribution is described by an order parameter which takes the values C_L = 1 and C_G = –1 in the liquid and gas-bulk phase, respectively. In a thin transition layer, the diffuse interface, C varies rapidly but smoothly. The temporal evolution of C is governed by the convective Cahn-Hilliard equation:

(2)

where u is the velocity field and M denotes the mobility. The chemical potential ϕ represents the variational derivative of free energy with respect to the order parameter. In the present study, it is given by:

(3)

Here, ε is the capillary width, which determines the thickness of the diffuse interface. For an equilibrium system, the mixing energy density λ and ε are related with interfacial tension σ as ⁴¹, ⁴⁴:

(4)

To account for surface wettability, various boundary conditions have been proposed for the phase-field method ⁴⁵. The approach in this paper follows ³⁶, where local equilibrium of free energy at the wall is expressed as:

(5)

Here, θ is the energy-equilibrium contact angle and n_s is the unit normal to the wall surface pointing outside the solid. Even though the boundary condition (5) incorporates the equilibrium contact angle, it recovers in combination with a no-slip condition at the solid wall the Cox spreading law ⁴⁶, ⁴⁷, which can be regarded as a representative dynamic contact angle model.

In the present study, both phases are assumed incompressible, immiscible, and Newtonian. The isothermal two-phase flow problem can thus be described by a divergence-free velocity field, Eq. 6, in combination with the single-field Navier-Stokes, Eq. 7:

(6)

(7)

Here, p denotes the pressure and g the gravity vector. The surface tension term in Eq. 7 is modeled as ³⁶, ⁴¹:

(8)

By this potential formulation, non-physical spurious velocities, which may occur near the interface in predominance of surface tension forces, are significantly reduced by several orders of magnitude ⁴⁸. The locally varying density and viscosity are computed from the order parameter field as:

(9)

The above set of governing equations is implemented in the open source C++ library OpenFOAM® (foam-ext 1.6) and solved by a finite-volume method (code phaseFieldFoam). Here, spatial derivatives are approximated by a second-order scheme with the Gauss linear correction while temporal derivatives are approximated by the Crank-Nicholson scheme. Adaptive time step control is used with the maximum Courant number limited to 0.1.

3.2 Computational Setup

For numerical efficiency and to allow for a large number of computational cases, all simulations are performed in a wedge-type geometry assuming rotational symmetry. Fig. 2 presents a sketch of the computational domain. The size in radial direction (r) is 6D₀, while the height in vertical direction (z) is 1.5D₀. The entire domain is discretized by a uniform mesh with identical grid size h in both directions.

The boundary conditions at the various borders of the computational domain are listed in Tab. 3. It has been checked that with the above radial size of the domain the sidewall is without influence on the spreading process.

The initial phase distribution corresponds to a spherical droplet (diameter D₀), which is in point-contact with the bottom wall; see Fig. 2. The initial order parameter field is already diffuse in accordance with the prescribed value of ε, cf. ⁴⁸. The gas phase is initially stagnant. For the liquid phase (C ≥ 0), the z-component of the velocity is initialized with value –U₀ to prescribe the experimental impact velocity of the droplet. Numerical studies in the context of a previous publication ³⁹ proved that initializing the droplet from a certain height above the surface instead from point-contact has only a small effect. While the advancing phase of spreading and the maximum spreading radius are only slightly affected, the late receding phase may be influenced by the formation of a gas bubble below the droplet, which may form in case the drop is initialized above the surface.

All physical properties in the simulations correspond to room temperature. The liquid-phase properties are taken from Tab. 1. The gas density is ρ_G = 1 kg m⁻³ and the gas viscosity is μ_G = 1.48 × 10⁻⁵ Pa s. The wetted wall is considered as atomically smooth having negligible hysteresis. In the wetting boundary condition, Eq. 5, the measured equilibrium contact angle θ = 50.3° is used. Gravity points in negative z-direction with gravitational acceleration g = 9.81 m s⁻².

Similar to ³³, the numerical parameters of the phase-field method are chosen as follows. The capillary width ε is obtained indirectly via the Cahn number Cn = ε/D₀, which is fixed to Cn = 0.01 in the present study. The mixing energy density is obtained from Eq. 4 with σ as given in Tab. 1. The Cahn-Hilliard mobility parameter is related to the capillary width as M = χε². The numerical value of χ is usually chosen on the order of 10⁻¹–10. Its magnitude has a slight influence on the spreading process and the maximum spreading radius ³⁹. For all simulations in the present study, χ = 0.5 m s kg⁻¹ is employed. Preliminary simulations showed that the maximum spreading radius varies by about approximately 4 % when χ is varied by one order of magnitude.

The diffuse interface is resolved by about four mesh cells, corresponding to h = ε. With Cn = 0.01, the latter criterion corresponds to a resolution of 100 mesh cells per initial drop diameter which is used for all present simulations. The typical CPU time for one test case on a single processor of a standard PC is about 20 h. For analysis of the numerical results, the interface location is identified by C = 0.

4.1 Droplet Morphology

Fig. 3 compares the instantaneous droplet shapes in experiment and simulations for one representative case (D₀ = 1.95 mm, U₀ = 1.95 m s⁻¹) at different time instants of the wetting process. Generally, the simulation results match well with the experimental counterparts in terms of the global morphology evolution of the droplet. At t = 0.3 ms, one observes the formation of a thin liquid rim at the droplet bottom while the spherical shape in the upper part of the droplet is still maintained. As the droplet spreads further out, it exhibits a flattened shape until the maximum spreading is reached at about t = 3.3 ms. Thereafter, the contact line recedes with the drop gradually recovering a spherical cap shape corresponding to the equilibrium state as shown for t = 1000 ms.

In spite of the qualitative agreement, there exist also some differences between experiment and simulation. At t = 1.3 ms, the fingering phenomena at the edge of the flattened droplet observed in the experiment cannot be reproduced by the rotationally symmetric simulation. For the recoiling stage, capillary waves occur at the free surface of the drop in the experiment (e.g., t = 8.3 and 10.3 ms). These arise due to the pinning of the contact line on the rough surface. In contrast, the gas-liquid interface in the simulation is almost flat as on the smooth numerical surface the droplet recoils without any pinning.

4.2 Spreading Dynamics

A typical measure to quantify wetting dynamics is the spreading factor (β) defined as the ratio between the instantaneous diameter of the wetted circular base area of the droplet and the initial droplet diameter. Fig. 4 illustrates the experimental and numerical time evolutions of β for both droplet sizes. Discrepancies for t < 1 ms can be attributed to the method of initialization in the simulations. For t = 0 s, the interface is initialized with a finite thickness resulting in an overprediction of the spreading factor in the first instants of droplet impact. In the experiments, precise determination of the instant of time of the first contact between droplet and solid surface is limited by the imaging frame rate. Accordingly, a slight shift of the experimental and numerical spreading data in time can be assumed.

In Fig. 4, some expected general trends could be observed in experiment and simulation. For a fixed droplet diameter, an increase of the impact velocity results in a larger maximum spreading factor (β_max) achieved at an earlier instant in time. After reaching the maximum spreading radius, the relaxation phase starts where the droplet recoils until it attains the equilibrium state, where β is constant in time. For the advancing phase, the agreement between experiment and simulation in general is good. The agreement in β_max is satisfactory for the two largest velocities of each drop size, but is less good for the lowest velocity. Concerning the recoil phase, the contact line recedes much faster in the simulations as compared to the experiment. Consequently, the droplet reaches the final static state in the simulation earlier as compared to the experiment.

The differences in the receding phase can be rationalized by the fact that in the simulation the solid surface is assumed perfectly smooth whereas in the experiment the surface has finite roughness. The latter causes frictional and pinning effects, which slow down the contact line motion during the receding phase. Therefore, in the experiment, the droplet shape is approaching the equilibrium state much slower and with a longer relaxation phase as compared to the simulations. In the experiment with lowest initial kinetic energy (D₀ = 1.95 mm, U₀ = 1.39 m s⁻¹), the receding phase is not finished within the measurement time as β is still decreasing in Fig. 4a. The described differences between the experimental and numerical results indicate that the surface roughness plays a notable role in the relaxation phase. In the advancing phase, where inertial effects dominate, surface roughness is less important in contrast. This behavior is in accordance with the experimental findings on the relaxation phase reported in ²⁰ and was also noted in our previous numerical study ³⁹.

The final equilibrium states in experiment and simulation agree quite well, with exception of the case just discussed (D₀ = 1.95 mm, U₀ = 1.39 m s⁻¹). The terminal values of β are almost independent on drop size and impact velocity. This is reasonable, given that the Eötvös number is of order one or below (cf. Tab. 2) so that gravitational forces are too small to enforce a notable deviation of the terminal drop shape from a spherical cap. In fact, the terminal values of β in Fig. 4 are in close agreement with the spreading factor of a drop forming a spherical cap ²⁰ given by

(10)

for θ = 50.3°. This demonstrates that the measured equilibrium contact angle is reasonably accurate.

4.3 Maximum Spreading Factor

The maximum spreading factor (β_max) is an important parameter for characterizing droplet impingement and wetting. This parameter is of particular interest for the application background of the present study, where coalescence of neighboring impinging droplets may initiate UWS film formation. Additionally, the size of wetted area affects the evaporative cooling of the wall. If β_max can be reduced, the probability of droplet interaction and UWS liquid film formation might decrease.

In literature, various correlations have been proposed for β_max ¹⁶, ¹⁷, ¹⁹, ²⁰, ⁴⁹. At present, no generally accepted universal correlation exists. Instead, the applicability of a correlation depends on droplet parameters and wall properties. Here, the empirical correlation of Scheller and Bousfield ²⁴ is considered which relates the maximum spreading factor with the dimensionless group ReWe^0.5 = Re²Oh as:

(11)

This relation is obtained by experiments for drops consisting of glycerol-water-ethanol mixtures with parameters in the following ranges: liquid viscosity 1–300 mPa s, surface tension 40–73 mN m⁻¹, contact angle 35–90°, drop diameter 2.0–4.0 mm, impact velocity 1.3–4.9 m s⁻¹. The ranges of the Reynolds and Ohnesorge numbers are 19 ≤ Re ≤ 16 400 and 0.002 ≤ Oh ≤ 0.58, respectively. Since all parameters in the present study are within or close to the respective ranges, Eq. 11 is especially suited to serve for a comparison here.

Bayer and Megaridis ⁵⁰ performed experiments with water droplets (D₀ = 1.4 mm, U₀ = 0.47–2.4 m s⁻¹, Oh = 0.003, We = 3–120) impacting on substrates with contact angles 20°, 74°, and 135°. By a regression fit, they obtained a correlation similar to Eq. 11 but with prefactor 0.72 and exponent 0.14. Fig. 5 compares both correlations. The correlation of Bayer and Megaridis ⁵⁰ is determined for a much smaller range of Re²Oh and is terminated at 2.7 × 10⁴, where fingering instabilities are expected to appear. This correlation predicts slightly smaller values of β_max as compared to ²⁴. The reason may be the incorporation of data for a hydrophobic surface, which results in lower values for β_max as compared to hydrophilic surfaces.

Fig. 5 compares the present experimental and numerical results for β_max (including cases not listed in Tab. 2) with the two latter correlations from literature. The agreement between the present experimental and numerical results can be considered good for the smaller drop, with a maximum deviation of about 6 %. For the larger drop, the deviation is higher and up to 11 %. No systematic over- or underestimation of the experimental values by the simulation can be noted. The agreement of the present results for β_max with Eq. 11 is good, considering the maximum error of the correlation being about 5 % ²⁴.

The four leftmost numerical data in Fig. 5 for droplet size D₀ = 1.95 mm are without experimental counterpart. The corresponding impact velocities of 0.15, 0.25, 0.5, and 1.0 m s⁻¹ are below the experimental range in ²⁴. For the two lowest impact velocities, the numerical values for β_max are quite large when compared with Eq. 11. However, this obvious disagreement is plausible. As U₀ decreases, β_max decreases and asymptotically approaches β_s-cap as values β_max < β_s-cap = 1.7425 cannot be reached for the present contact angle θ = 50.3°.

4.4 Perspective for UWS Droplets in Technical Application

In automotive SCR applications, droplets are usually much smaller than investigated in the present experiments. For pressure-driven spray injectors, the Sauter mean diameter is in the range of 60–80 µm with wall-normal droplet impact velocities up to 20 m s^{−1 6}. Therefore, it is of interest in how far Eq. 11 applies to such conditions as well. To investigate this topic, one additional simulation for a much smaller droplet size with higher impact velocity is performed keeping the other parameters unchanged. The droplet size and impact velocity are chosen as D₀ = 0.195 mm and U₀ = 10.97 m s⁻¹, respectively, corresponding to Re = 1802, We = 340, and Re²Oh = 33216. The latter value is the same as in Fig. 3 where D₀ = 1.95 mm and U₀ = 1.95 m s⁻¹. Due to the higher impact speed, the spreading of the 0.195 mm droplet is much faster as compared to Fig. 3.

Fig. 6 displays a visualization of the droplet shape during the early spreading process at two instants in time. At 7 µs, a lift-off of the lamella can be seen in Fig. 6a whereas at 11 µs a contact of the lamella with the solid wall ahead of the original contact line is visible in Fig. 6b. Both images are very similar to the sketch in Fig. 2a of Thoroddsen et al. ⁵¹, who performed experiments with water-glycerin drops of much larger size (D₀ ≈ 5.1 mm). The authors observed the tip of the spreading lamella being separated from the surface and riding on a cushion of an air layer (corresponding to present Fig. 6a). This is followed by a localized wall contact ahead of the initial contact line (corresponding to present Fig. 6b) leading to bubble entrapment while the lamella front becomes unstable due to fingering instabilities ultimately resulting in splashing. The present numerical results in Fig. 6 thus seem to be physical. However, the simulation is not continued in time as fingering instabilities and splashing cannot be captured by the assumed rotational symmetry.

Though the group Re²Oh = 33216 is identical for the simulations in Figs. 3 and 6, the spreading behavior in both cases is very different as the larger drop deposits while the smaller one splashes. While the spreading correlation in Eq. 11 is valid for UWS drops of millimeter size, it is obviously not applicable for much smaller droplets as they occur in sprays for automotive SCR applications.