Application of dimensional analysis in estimating the wall film thickness created by a liquid spray impact

This study presents a theoretical investigation on the formation and spreading of a liquid film on a flat and rigid wall due to spray impact. The new model for predicting the average film thickness is a function of mean Reynolds number of the impacting drops, flux density of the impacting droplets, and the average drop diameter. The theoretical derivation exhibits good agreement with the measured data in the thin film condition. Experimentally the average film thickness accumulated on the wall and the spray has been measured using a high-speed CCD camera and a dual-mode phase Doppler instrument, respectively. Introduction With the process of spray/wall interaction, it has become apparent in recent years that the liquid film formed on the wall plays an important role in determining the velocity and size of ejected droplets as well as the deposited mass fraction, see e.g., Bai et al. (2002), Cossali et al. (1999). Nevertheless, the formation of a wall film is often neglected in spray impact models, although prediction of average film thickness and average velocity is very important for many industrial applications, especially for those involving spray cooling systems or for fuel injection sprays onto heated walls, since the wall film significantly affects the efficiency of heat transfer on the surface. In some applications, it is desirable to eliminate the deposited film on the wall as far as possible, e.g. in internal combustion engines, whereas in other cases a maximum deposition is required, e.g. in spray coating, spray painting or agricultural sprayers. On the other hand, the induced fluctuations in the liquid layer formed on the rigid walls may decrease the quality of coated or painted surfaces. The average film thickness can also affect the properties of the secondary spray, splashing threshold, ejected mass and number of secondary droplets. Numerous models exist regarding the formation of the wall film generated by an impacting liquid spray, see e.g., Stanton and Rutland (1996), Lee et al. (2001), and Bai and Gosman (1996), Ahmadi-Befuri et al. (1996), Kalantari and Tropea (2006e). The model of Stanton and Rutland (1996) solves the continuity and momentum equations for a 2-D film flow over an arbitrary solid surface using the Euler method. This model considers many physical effects such as shear forces and dynamic pressure of impacting droplets, but neglects the Laplace (capillary) pressure arising from curvature of the air-liquid film interface. In the present study, a new expression is given based on the dimensional analysis for predicting the average film thickness accumulated on the wall due to a liquid spray impact. The theoretical derivation for the average film thickness then compared with the measured data in the thin film condition, i.e., (h d10b < 1, where h is the average film thickness and d10b is the mean drop diameter before impact). Theoretical approach In the impingement region of an inertial spray, the average film thickness created on the wall depends on the several parameters of the impacting spray ; normal and tangential component of impact velocity u b and vb , volume flux density of impacting spray (q = q A ; “q” and “A” to be volume flux of the impacting spray(m/s) and the reference area over which flux is measured), volume-averaged diameter of impacting droplets (d30b) defined by d30b = di 3 N i=1 1/3 , density (L) and dynamic viscosity of the liquid (μ), as well as the boundary condition of the target (e.g., flat or curved target surface or target surface completely covering with spray); average target surface roughness (ε ) and target size (D). A general expression for the average film thickness can be written as h = Ψ u b , v b , d30b , q b , ρL , μ, ε , Dspray , D (1) * Corresponding author: dkalantari2000@yahoo.com 12 ICLASS 2012 Abbreviated Paper Title Abbreviated Paper Title Abbreviated Paper Title Abbreviated Paper Title 2 where, Dspray is the diameter of the effective impinging spray on the target defined as: Dspray = 2xNozzle tan α 2 , α is the spray cone angle. The parameters u b , vb , qb , d30b and Dspray vary with nozzle pressure and nozzle height above the target. The three first parameters can be combined into an impact Reynolds or Weber number depending on the dimensional analysis output. Volume flux of impacting spray (q) depends on the atomizing pressure (P) with a power law; q ∝ P (Bernoulli’s equation). For characterizing the average film thickness accumulated on the wall due to a normal spray impact (negligible tangential velocity component of the impacting droplets) and for a spray covering the target surface (i.e., Dspray/D>1), variables involving in the film thickness (1) can be reduced to h = Ψ u , d30b , q b , ρL , μ, ε (2) Using the dimensional analysis with considering the d30b, u b and ρl as repeating variables, the П-numbers can be written as Π1 = d30b a1 u b b1 ρl h = MLT Π2 = d30b a2 u b b2 ρl μ = MLT Π3 = d30b a3 u b b3 ρl q b = M LT Π4 = d30b a4 u b b4 ρl ε = MLT (3) Equating the exponents of two sides of each П-number and Solving explicitly the obtained equations for a1, b1, . , c4, one obtains Π1 = d30b h Π2 = d30b −1 u b −1 ρl μ Π3 = u b q b Π4 = d30b ε (4) Therefore based on the П-theorem, the functional relationship for the obtained П-numbers can be written in the form of h d30b = f 1 Reb , q b