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Turbulent mixing of oil droplets in a round water jet

In order to f**in**d the response time, one should be able to predict the motion **of** a droplet **in** anaccelerat**in**g fluid. However, deriv**in**g 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 **in**teract with eddies and maylose their **in**itial shape. Therefore, it is only relevant to consider the Stokes number if the **droplets**are stable (see section 2.2.3).To estimate the response time **of** a droplet **in** an accelerat**in**g 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 sur**round****in**g fluid. This drag force is equal to [Batchelor, 1967]:( µ +32F drag = 2πd d µµ )dU d (2.48)µ + µ d**in** which µ d is the dynamic viscosity **of** the dispersed phase and U d is the velocity **of** the dropletrelative to the sur**round****in**g fluid. In this expression, the effects **of** **in**ternal circulation with**in** thedroplet are taken **in**to account. The size **of** the drag force determ**in**es the time that is neededto respond to accelerations **of** the sur**round****in**g fluid. The higher the drag force, the faster theresponse. When the simplified case is considered that a particle (plus added mass) with an **in**itialslip 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 d**in** 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 fluidsur**round****in**g the particle follows:τ particle = d2 d ρ d18µ(2.52)which is a common approximation for calculat**in**g 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 f**in**ite. 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 toma**in**ta**in** 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 sur**round****in**g fluid, they tend to risewhen issued **in** a vertical set-up. A parameter that gives an **in**dication **of** the size **of** the buoyancyeffect, is the term**in**al velocity **of** a ris**in**g droplet **in** a stagnant fluid. The term**in**al 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 term**in**al 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 be**in**g 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 sur**round****in**g fluid, may, under certa**in** circumstances,deform and eventually break up. Inertia forces which are **in**duced by the fluctuations have adestabilis**in**g effect on the **droplets**, whereas the **in**terfacial 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 **in**terfacial 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 f**in**d the velocity fluctuations over the size **of** a droplet the follow**in**g relation is applied [Walstra,1993], which is derived for the **in**ertial field **of** isotropic turbulence.u ′ d = C(ɛd d ) 1/3 (2.58)**in** which C is a constant **of** order unity. Substitut**in**g the expression for the velocity fluctuations**in** the Weber number, yields:W e d = ρ dɛ 2/3 d 5/3d(2.59)σThe dissipation rate, ɛ, can be estimated accord**in**g to equation 2.45. At large Weber numbers,the **in**ertia forces dom**in**ate and break-up is likely. If W e ≪ 1 the droplet ma**in**ta**in**s 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 s**in**gle 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

- Page 1: Turbulent mixing of oil dropletsin
- Page 4 and 5: ContentsSamenvattingSummaryviviii1
- Page 7 and 8: SamenvattingDit project richt zich
- Page 9: SummaryThis project focuses on the
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- Page 14 and 15: 2.1.1 Boundary-layer equationsStart
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- Page 20 and 21: 0.020.015K−theoryPrandtlfrom Gaus
- Page 22 and 23: 2.1.6 Kolmogorov scalesVia a cascad
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- Page 28 and 29: conserved.The centreline concentrat
- Page 30 and 31: Figure 3.1: Experimental set-up.3.1
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- Page 34 and 35: Figure 3.6: Composition of peaks in
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- Page 43 and 44: Table 4.2: Characteristics of dropl
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15measured valuesself−similarity

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0.030.02measurementstheoretical pro

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0.15−r∂U/∂z∂(rV)/∂rresidu

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taken into account. Also, a theoret

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10510095z [mm]90858075−30 −20

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90PIVWareDaVis8580U c[mm/s]75706590

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0.2PIVWareDaVis0.15V rms/U c0.10.05

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List of Figures2.1 A turbulent jet,

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List of SymbolsRomana - velocity of

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Greekα - spreading rate of the jet

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ReferencesAanen, L. (2002), Measure