Phenomenological models of cavitation erosion kinetics

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4 International Cavitation Erosion Test Seminar, Sopot, 1-2 June 2000where the expression under the integral sign denotes erosion rate at the surface element uncoveredin the instant T.F.J.Heymann states that realistic erosion curves can be obtained by assuming the f(t)function to vanish in the initial period of exposure (t ≤ T 0 ), and then to be described by athree-parameter logarithmic-normal distribution[ ln( t − T ) − m]21 ⎪⎧−⎪⎫0f () t = exp⎨2 ⎬(2)σ ( t − T0 ) 2π⎪⎩ 2σ⎪⎭For the original surface layer F.J.Heymann proposes to use a distribution with parametersdifferent from those applied for the subsequent layers. However, there are no hints on themethod of their determination to be found in reference [4].Later on, J.Noskievič [5] and K.Steller [6] proposed models of erosion kinetics referringdirectly to the erosion curve pattern.J.Noskievič has noticed that the instantaneous erosion rate curve, IER =IER(t) remindsthat of damped harmonic oscillator motion. The equation of motion of such an oscillator canbe written asd2( IER) d( IER)dt22+ 2α + β IER = I . (3)dtwith α and β denoting coefficients responsible for material properties (α - work-hardeningcapability, β - cavitation resistance), and I - parameter characterising intensity of cavitationerosive activity (I = const). By assuming erosion rate and its first derivative to vanish at thebeginning of the process, the solution of equation (3) can be written down as−κtκt[( α −κ) e − ( α + κ ) e ]−αt⎧ e⎪vs+ vs⎪2κ−αt⎪vs− vs( 1+α2t)eIER()t = ⎨⎪ ⎛ α ⎞vs− vs⎜cosωt− sinωt⎟e⎪ ⎝ ω ⎠⎪⎩vs− vscosωt−αtififififα > βα = ± βα < βα = 0andα ≠ 0, (4)2 22where κ = α − β and ω = β −α2 , whence the volume loss curve is given by:∆V= vs−αt⎡ α e ⎛α+ κ kt α −κ−kt⎞⎤⎢t− 2 + ⎜ e − e ⎟⎥⎣ β 2if ⏐α⏐ > β (5a)2κ⎝α−κα + κ ⎠⎦∆ V= vs⎡ 2 ⎛ 2 ⎞⎢t− + ⎜t+ ⎟e⎣ α ⎝ α ⎠−αt⎤⎥⎦if α =±β(5b)⎧ α α t ⎡⎛⎞⎤⎫−αα ω∆V = vs⎨t− 2 2+ 2e ⎢⎜− ⎟ sin ωt+2 cosωt⎣⎝⎠⎥⎬,⎩ β β ω α⎦⎭if ⏐α ⏐< β and α ≠ 0 (5c)⎞⎜1∆V= v − s t ω t ⎟⎝ ω sin ,⎠if α = 0 (5d)
J.Steller, P.Kaczmarzyk: Phenomenological models of cavitation erosion progress. 5The formula proposed by K.Steller is based on the notion of cavitation resistance R cav ,clearly linked to that of A.P.Thiruvengadam's erosion strength [7] and defined by the equation:Pt = R cav ∆V (6)where P is the power used to erode the material subject to cavitation impingement. Cavitationis assumed a continuous process (proceeding at variable speed) resulting in the change of bothmaterial surface and its internal structure, responsible for cavitation resistance. Cavitationresistance dependence on duration of cavitation impingement is assumed to be described byan exponentially decreasing functionThe formula derivedRcav( 1+χ )−κtχ + e=1 R 0χ +Pt2 1≈σ∆V=−κt2R0( χ + e )( + χ )EPt−κt( χ + e )shows clear dependence of the volume loss curve ∆V = ∆V(t) on some basic material parameters(modulus of elasticity E and ultimate stress values σ), χ factor determining the ratio ofultimate and initial cavitation resistance (equation (7)), κ parameter defining the speed ofcavitation resistance decrease in course of the process and power P used to erode the material.In mid eighties L.Sitnik [8] assumed the time interval ∆t between removal of the consecutiveparticles from the material surface to be a random function with a three-parameterlogarithmic-natural distribution functionThe volume loss curve derived takes the formFpβ p{ }( ∆t) = 1 − − α [ ln ( ∆t∆t0 + 1)]p(7)(8)exp . (9)β p[ ( t ∆t∆V= α ∆Vln 1)]p0 0 +with ∆V 0 denoting certain characteristic volume value. The α p parameter is characteristic forthe intensity of erosive action of the phenomenon. It is worthwhile to notice that the model isa simplified version of the F.J.Heymann model.(10)3. DETERMINING THE VOLUME LOSS CURVE PARAMETERSACCORDING TO THE SELECTED MODELSOF CAVITATION EROSION KINETICSThe least squares method has been adopted in order to determine the volume loss parametersaccording to selected models. The task can be reduced to that of minimising the expression2[=∑α∆V−χ ( k)U kjjj( , t )]with α j denoting the weight coefficient, U( . , t) - a function describing erosion progress accordingto the model selected, and k - a set (vector) of parameters to be determined by meansof the minimising procedure.During calculations of that kind particular attention should be brought to an appropriateselection of the zero-th approximation.j2,(11)
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Phenomenological models of cavitation erosion kinetics

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