INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

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38 2. Mathematical Modeling solid layer thickness, δ, as where mC pL dT s dt = Q L + ṁL V (T s ) 1 + Ñu k gfδ/[2k s (R − δ)] − ṁL V (T s ), (2.59) Ñu is the modified Nusselt number defined by Eq. (2.49), which accounts for the effect of convective droplet heating [62], k s and k gf are the thermal conductivity of the solid layer and in the film, respectively, and Q L is the net heat transferred to the droplet [62], given by Eq. (2.58). Similar to Eq. (2.55), the second term in denominator inside the bracket of Eq. (2.59) denotes the resistance due to solid formation at the droplet surface, and its effect becomes significant only when the solid layer thickness, δ, is positive. The difference between heat transfer and mass transfer resistance is that the ratio of diffusion coefficients D f /D s is larger than the ratio k g /k s [151], which implies that resistance to mass transfer due to solid layer formation is higher than the heat transfer. In the present study, simulations are also carried out with rapid mixing model (RMM) which is a simple model based on the assumption that the liquid mixture inside the droplet is always homogeneous (no spatial gradients of mass fraction within the droplet) and infinity conductivity within the droplet, thus the droplet is at uniform temperature at every time. In this work, the RMM is extended to account for solid layer resistance on the droplet evaporation rate and heating, thus the governing equations in the RMM are Eq. (2.55) and Eq. (2.59), and the time evolution of solute (i = 2) mass fraction is calculated based on the simple mass balance of solute and solvent mass fraction within droplet, given as Y 2,RMM = Y 02 − m 02 m − ṁ , (2.60) where Y 02 and m 02 are the initial solute mass fraction and solute mass within the droplet, respectively. 2.4.2 Droplet Motion The dynamics of liquid droplets in sprays is the basic physical process that needs to be computed for the coupling of gas–liquid phases due to its strong dependance on the flow of surrounding gas. The droplet velocity v at position x can be computed as following v = dx dt . (2.61) The acceleration of droplets due to different forces acting on droplets can be written as [174] dv dt = ΣF + ρ ( g Du 2ρ l Dt − dv ) , (2.62) dt
2.4. Single Droplet Modeling 39 where the first term in R.H.S includes all the forces such as aerodynamic drag, gravity, Basset, lift, and buoyancy etc. and the second term in R.H.S is the added mass D force [174]. In Eq. (2.62), is the substantial or material <strong>der</strong>ivative. Dt The force experienced by the droplets due to difference in velocities of droplets and surrounding gas is known as drag force, F d . The droplet velocity evolution by interactive drag induced by the surrounding gas, and gravity per unit droplet mass is commuted using the following relation, which describes droplet motion [175] F d = 3 8 1 ρ g (u − v)|u − v|C D + g, (2.63) r ρ l where ρ g and u are the density and velocity of the surrounding gas, respectively, while ρ l , C D and g are liquid density, drag coefficient, and gravitational acceleration, respectively. The dependencies of the drag force are confined to the droplet radius, droplet shape, droplet density, ρ l , relative velocity between gas and droplet, u − v, gas density, ρ g , kinematic viscosity of the gas, η g , and surface tension, σ d . The drag coefficient, C D , is calculated as a function of the droplet Reynolds number, Re d = 2rρ g |u − v|/µ f , where µ f is the mean dynamic viscosity in the film, as [176] { 24 (1+ Re C D = 1 d 6 Re0.687 d ) if Re d
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INAUGURAL-DISSERTATION zur Erlangun
Page 5:
Dreams can never become a reality w
Page 8 and 9: II ial direction). Earlier studies
Page 10 and 11: IV DQMOM wird die Tropfengrößen-
Page 13 and 14: Contents Abstract . . . . . . . . .
Page 15 and 16: List of Tables 4.1 Initial weights
Page 17 and 18: List of Figures 1.1 A schematic dia
Page 19 and 20: List of Figures XIII 4.27 Effect of
Page 21 and 22: 1. Introduction Drying has found ap
Page 23 and 24: 3 Fig. 1.2: Chemical structure of p
Page 25 and 26: 5 via inhalation aerosols. Dry powd
Page 27 and 28: 7 equations are written in terms of
Page 29 and 30: 9 for through appropriate sub-model
Page 31 and 32: 2. Mathematical Modeling Spray cons
Page 33 and 34: 2.1. State of the Art 13 the govern
Page 35 and 36: 2.1. State of the Art 15 and if the
Page 37 and 38: 2.2. Euler - Lagrangian Approach 17
Page 39 and 40: 2.2. Euler - Lagrangian Approach 19
Page 41 and 42: 2.3. Euler - Euler Approach 21 quan
Page 43 and 44: 2.3. Euler - Euler Approach 23 Will
Page 45 and 46: 2.3. Euler - Euler Approach 25 of t
Page 47 and 48: 2.3. Euler - Euler Approach 27 With
Page 49 and 50: 2.4. Single Droplet Modeling 29 col
Page 51 and 52: 2.4. Single Droplet Modeling 31 the
Page 53 and 54: 2.4. Single Droplet Modeling 33 pol
Page 55 and 56: 2.4. Single Droplet Modeling 35 gra
Page 57: 2.4. Single Droplet Modeling 37 thr
Page 61 and 62: 2.4. Single Droplet Modeling 41 (a)
Page 63 and 64: 2.4. Single Droplet Modeling 43 not
Page 65 and 66: 3. Numerical Methods In the numeric
Page 67 and 68: 3.2. Spray Modeling 47 and solid la
Page 69 and 70: 3.2. Spray Modeling 49 ⎛ ⎞ ⎛
Page 71 and 72: 3.2. Spray Modeling 51 application
Page 73 and 74: 3.3. Numerical Performance 53 where
Page 75 and 76: 4. Results and Discussion Results p
Page 77 and 78: 4.1. One-dimensional Evaporating Wa
Page 87 and 88: 4.2. Two-dimensional Evaporating Wa
Page 97 and 98: 4.3. Single Bi-component Droplet Ev
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4.3. Single Bi-component Droplet Ev
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4.3. Single Bi-component Droplet Ev
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4.4. Two-dimensional Evaporating PV
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5. Conclusions and Future Work The
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101 proved UNIFAC-vdW-FV method for
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Appendix
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106 A. Nomenclature Symbol Unit Des
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108 A. Nomenclature S n S g S l Vec
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110 A. Nomenclature List of Abbrevi
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Bibliography [1] K. Masters: Spray
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BIBLIOGRAPHY iii [24] K. D. Squires
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BIBLIOGRAPHY v [49] D. L. Marchisio
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BIBLIOGRAPHY vii [72] G. M. Faeth:
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BIBLIOGRAPHY ix [95] C. Cercignani,
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BIBLIOGRAPHY xi [119] R. O. Fox: Hi
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BIBLIOGRAPHY xiii [142] R. Clift, J
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BIBLIOGRAPHY xv [170] G. Brenn, L.
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BIBLIOGRAPHY xvii [197] M. Stieß:
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BIBLIOGRAPHY xix [222] I. S. Grigor
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