- Text
- Velocity,
- Turbulent,
- Concentration,
- Droplets,
- Equation,
- Axial,
- Radial,
- Profiles,
- Droplet,
- Fluid,
- Mixing

Turbulent mixing of oil droplets in a round water jet

If two **droplets** collide, dur**in**g the **in**teraction time, t **in**t , the fluid film between the **droplets** mustvanish, dur**in**g the dra**in**age time, t dra**in** . The probability **of** coalescence **in** case **of** a collisiondepends on the ratio **of** these time scales:P coal = e − t dra**in**t **in**t (2.61)It is beyond the scope **of** this project to give a complete quantitative analysis **of** this coalescenceprocess so this probability is simply set to 1. In order to predict whether coalescence might playa role **of** importance, the collision frequency **of** **droplets** per unit volume, n c ω col , with**in** the microscale **of** size η k is compared to the reciprocal time the **droplets** are **in** this micro structure, 1/τ k .Then, if 1/τ k ≫ n c ω col ηk 3 , the effects **of** collisions are neglected. This statement can be rewrittenas:c 2 (η k /d d ) 3 ≪ 1 (2.62)2.2.4 Concentration development **in** a turbulent **jet**In this section, the **oil** **droplets** are assumed to behave as a passive scalar **in** a turbulent flow fieldwithout diffusion. After Reynolds decomposition **of** the **oil** concentration (c = c+c ′ ), the averagedtransport equation reads:∂ū¯c∂z + 1 ∂r¯v¯c= − ∂u′ c ′r ∂r ∂z− 1 ∂rv ′ c ′(2.63)r ∂rThe mean concentration scales with c s , while the turbulent transport terms u ′ c ′ and v ′ c ′ scalewith UC. It will be verified experimentally that the ratio **of** these scales is determ**in**ed by:UC l)= O((2.64)u s c s LApply**in**g the boundary-layer approximation (2.5) and the cont**in**uity equation (2.3), the equation**of** transport can consequently be written as:u ∂c∂z + v ∂c∂r = −1 ∂rv ′ c ′r ∂r(2.65)The term v ′ c ′ can be **in**terpreted as the turbulent transport **of** scalar concentration **in** the r-direction (turbulent flux).Like the velocity pr**of**iles, the concentration pr**of**iles are assumed to be self-similar as well. Moreover,it is assumed that the typical length scale **of** the concentration pr**of**ile, l c , scales with thewidth **of** the velocity pr**of**ile: l c = γl. Consequently, the concentration pr**of**ile can be expressed asa function **of** η as well.c = c s h(η)v ′ c ′ = UCk(η) (2.66)**in** which c s it the concentration at the centrel**in**e **of** the **jet**. Now, the conservation **of** solute matterflux, Q, is considered:Q = 2π∫ ∞0∫ ∞rū¯cdr = 2πu s c s l 2 ηf(η)h(η)dη = Q 0 = c 0 Φ **jet** (2.67)**in** which Q 0 is the **in**itial solute matter flux def**in**ed by the **jet** flow, Φ **jet** , and the **in**itial concentration,c 0 . Us**in**g u s l = constant, it follows that, to ensure self-similarity, also c s l must be constant.Thus, the development **of** the centrel**in**e concentration as a function **of** z becomes:c s = c 2l = c 2α(z − z 0 )016(2.68)

Substitut**in**g (2.66) **in** the equation **of** transport (2.65) and us**in**g (1/c s )dc s /dz = −(1/l)dl/dz, itfollows that:Uc s dCu s dη F (η)h(η) = d ηk(η) (2.69)dηA self-similar solution is only possible if:Uc sCu s= c 3 = constant (2.70)Integrat**in**g (2.69) to η, gives the follow**in**g relation between the self-similar concentration andvelocity pr**of**iles. The boundary condition F(0)=0 is used to determ**in**e the **in**tegration constant.F (η)h(η) = 1 c 3ηk(η) (2.71)In order to f**in**d a solution for the concentration pr**of**ile h(η) it is necessary to formulate a closurehypothesis for the turbulent flux, v ′ c ′ . Aga**in**, K-theory is applied [H**in**ze, 1975]:v ′ c ′ ∂c≡ UCk(η) = −K c∂r = −K cc s dh(η) (2.72)l dηIn analogy with the eddy-viscosity, K, the eddy-diffusivity, K c , scales with the macro structure **of**the turbulence: K c = κUL. The unknown transport coefficient, κ, that is **in**troduced, determ**in**esthe ratio between matter transport and momentum transport.Substitut**in**g k(η) **in** the equation **of** transport (2.69) and tak**in**g K c = κu s Bαl (see section 2.1.5),it follows that:F (η)h(η) = −κBη d h(η) (2.73)dηUs**in**g the expression for F (η) derived **in** section 2.1.3, the solution for c s h(η) is given by:c ≡ c s h(η) = c s1(1 + η 2 /8B) 2/κ (2.74)Note, that if κ = 1, the concentration pr**of**ile and the velocity pr**of**ile collapse. The larger thistransport coefficient, the wider the concentration pr**of**ile becomes. In figure 2.7a the concentrationpr**of**ile is plotted for two values **of** κ.From the expression for the concentration pr**of**ile it is possible to derive the spread**in**g rate **of** theconcentration, α c , as a function **of** κ. The characteristic length scale, l c , is def**in**ed as the widthfor which the concentration is equal to 1 e c s. Say η = γ = l c /l, then the solution for h(η) reads:1e = 1(2.75)(1 + γ 2 /8B) 2/κThe relation between the spread**in**g coefficient, γ, and the transport coefficient, κ, becomes:√κ = 2ln(1 + γ 2 /8B) or γ = 8B(e −1/2κ − 1) (2.76)The spread**in**g coefficient, γ must be determ**in**ed experimentally. It is known from previous experimentsthat γ > 1, so the scalar transport is faster than the momentum transfer. In terms **of** κ,κ ≈ 1.4 [H**in**ze, 1975]. This could be expla**in**ed by the fact that the fluid carries the **oil** **droplets**,while its k**in**etic energy is dissipated and momentum is lost whereas the scalar quantity is always17

- 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
- Page 12 and 13: measurements such as Hot Wire Anemo
- Page 14 and 15: 2.1.1 Boundary-layer equationsStart
- Page 16 and 17: Consequently, a self-similar soluti
- Page 18 and 19: is a function of z, so F must be eq
- Page 20 and 21: 0.020.015K−theoryPrandtlfrom Gaus
- Page 22 and 23: 2.1.6 Kolmogorov scalesVia a cascad
- Page 24 and 25: In order to find the response time,
- Page 28 and 29: conserved.The centreline concentrat
- Page 30 and 31: Figure 3.1: Experimental set-up.3.1
- Page 32 and 33: water droplet at the outlet of a ca
- Page 34 and 35: Figure 3.6: Composition of peaks in
- Page 36 and 37: diameter of a 20 µm droplet is 4.7
- Page 38 and 39: presence of a highly seeded jet lea
- Page 41 and 42: Chapter 4ResultsThis chapter gives
- Page 43 and 44: Table 4.2: Characteristics of dropl
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- Page 47 and 48: 12.5measured valuesself−similarit
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- Page 57 and 58: 0.50.45measurementspolynomial fitAa
- Page 59 and 60: 1413.5measured valueslinear fit1312
- Page 61 and 62: 1measurementsGaussian fittheoretica
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- Page 69 and 70: Appendix ASingle-phase jet experime
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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