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Turbulent mixing of oil droplets in a round water jet

Turbulent mixing of oil droplets in a round water jet

In order to

In order to find the response time, one should be able to predict the motion of a droplet in anaccelerating fluid. However, deriving the equation of motion of a droplet in a turbulent flow field,is very complicated. See for example [Maxey, 1983], who derived an equation of motion for a rigidsphere. Moreover, if the droplets become too large, they start to interact with eddies and maylose their initial shape. Therefore, it is only relevant to consider the Stokes number if the dropletsare stable (see section 2.2.3).To estimate the response time of a droplet in an accelerating fluid, a very simplified case isconsidered: a spherical droplet with diameter, d d , in translational motion on which a drag force,F drag , is exerted by the surrounding fluid. This drag force is equal to [Batchelor, 1967]:( µ +32F drag = 2πd d µµ )dU d (2.48)µ + µ din which µ d is the dynamic viscosity of the dispersed phase and U d is the velocity of the dropletrelative to the surrounding fluid. In this expression, the effects of internal circulation within thedroplet are taken into account. The size of the drag force determines the time that is neededto respond to accelerations of the surrounding fluid. The higher the drag force, the faster theresponse. When the simplified case is considered that a particle (plus added mass) with an initialslip velocity U and mass m d is decelerated by the drag force, the equation of motion reads:(m d + 1 2 m f ) dU ddt= 1 6 πd3 d(ρ d + 1 2 ρ)dU ddt( µ +32= −2πd d µµ )dU d (2.49)µ + µ din which the added mass for a fluid sphere is taken as half the mass of the displaced fluid [Clift,1978]. The solution of this differential equation is as follows:U d = Ue − tτ d (2.50)in which the response time, τ d , is the characteristic time scale of the exponential decay:τ d = d2 d12µ (ρ d + 1 ( µ +2 ρ) µd)µ + 3 2 µ d(2.51)In the limit of µ d → ∞ and ρ d ≫ ρ, the response time of a solid sphere in a low density fluidsurrounding the particle follows:τ particle = d2 d ρ d18µ(2.52)which is a common approximation for calculating response times of particles [Raffel, 1998]. Equation2.51 will be used here, as the droplet density is close to the water density and the dropletviscosity is of course finite. Note, that in the calculation of the drag force, it is assumed thatthe droplet Reynolds number (Re d = U d d d /ν) is very small. Besides, the droplet is supposed tomaintain its spherical shape. In section 2.2.3 this aspect will be considered.2.2.2 BuoyancyIf droplets have a density that is lower than the density of the surrounding fluid, they tend to risewhen issued in a vertical set-up. A parameter that gives an indication of the size of the buoyancyeffect, is the terminal velocity of a rising droplet in a stagnant fluid. The terminal velocity, U term ,is reached if the buoyancy force, F buoy , is equal to the drag force on the droplet (see equation2.48):F buoy = 1 6 (ρ d − ρ)gπd 3 d = F drag (2.53)14

Equilibrium is reached if:U term = 1 d 2 d g ( µ +12 µ (ρ µd)d − ρ)µ + 3 2 µ d(2.54)If the terminal velocity is much smaller than the axial velocity in the jet, it is assumed that thebuoyancy effect is negligible.An alternative approach is to consider the acceleration of a plume of oil droplets in water, a plume .This acceleration is equal to:a plume =F bouy=(ρ d − ρ)cgm plume ρ d c + ρ(1 − c)(2.55)in which c is the volume concentration of oil droplets. Now, the acceleration of the plume canbe compared to the deceleration of the jet after being issued from the orifice, in order to get aquantitative measure of the buoyancy effect.2.2.3 Break-up and coalescenceA droplet which is subject to fluctuations of a surrounding fluid, may, under certain circumstances,deform and eventually break up. Inertia forces which are induced by the fluctuations have adestabilising effect on the droplets, whereas the interfacial tension force has a tendency to leavethe droplets spherical. The ratio between these forces is known as the droplet Weber number:W e d = ρ d(u ′ d )2 d dσ(2.56)in which σ is the interfacial tension and u ′ dthe typical velocity difference over the droplet withsize d d . This velocity difference is estimated with the Kolmogorov velocity scale (see equation2.43). The velocity fluctuation at the Kolmogorov scale, v k , as a function of the length scale, η kbecomes:v k = (ɛη k ) 1/3 (2.57)To find the velocity fluctuations over the size of a droplet the following relation is applied [Walstra,1993], which is derived for the inertial field of isotropic turbulence.u ′ d = C(ɛd d ) 1/3 (2.58)in which C is a constant of order unity. Substituting the expression for the velocity fluctuationsin the Weber number, yields:W e d = ρ dɛ 2/3 d 5/3d(2.59)σThe dissipation rate, ɛ, can be estimated according to equation 2.45. At large Weber numbers,the inertia forces dominate and break-up is likely. If W e ≪ 1 the droplet maintains its sphericalshape.Another phenomenon that may occur is coalescence of droplets. The probability of coalescencedepends on the collision frequency of droplets and the probability of coalescence in case of acollision. The collision frequency, ω col , for a single droplet smaller than the Kolmogorov lengthscale, depends on the number concentration of oil droplets, n c , and the time scale, τ k , as follows[Saffman, 1956]:ω col ∼ c ( ɛ) 1/2nc≈ d 3 d (2.60)τ k ν15

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