INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

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30 2. Mathematical Modeling liquid and film properties. The work of Sirignano [140] classified the single component droplet evaporation models into six types, and they are given in the order of complexity as, (1) constant droplet temperature model (also known as the d 2 law), (2) infinite liquid conductivity model (uniform but time dependent droplet temperature), (3) conduction limit (spherically symmetric transient droplet heating) model, (4) effective conductivity model, (5) vortex model of droplet heating, and (6) Navier – Stokes solution. There are various differences among these models, and some of these models are shown to be limits of another model [140]. Recent study of Sazhin [147] gives an overview of all the existing droplet evaporation models, particularly in the field of combustion studies. For evaporating water spray flow in air, the spatial gradients of the temperature within the droplet will not be significant when the evaporation conditions are room temperature and atmospheric pressure. Thus, in the current study, uniform but time dependent droplet temperature with convective effects can be used to predict the evaporation rate, and the droplet size regression. Therefore, for evaporating water sprays under room temperature and pressure conditions, the model of Abramzon and Sirignano [62] is implemented, which is a uniform temperature model that includes the convective effects, and considers the variable liquid and film properties. Here film means a thin layer across the droplet surface where the saturation of liquid vapor exists, and this vapor mass fraction is computed based on the vapor-liquid equilibrium. The rate of change of droplet mass with time due to convective evaporation and droplet heating in water spray is computed as [62] ṁ = 2πRρ f D f ˜Sh ln(1 + BM ), (2.41) where R is the droplet radius, ρ f is the density in the film, D f is the water diffusivity in the film, ˜Sh is the modified Sherwood number that accounts for the convective effects of droplet evaporation [62], given as ˜Sh = 2 + Sh − 2 B M (1 + B M ) ln(1 + B M ). (2.42) Here, Sh is the Sherwood number, which is defined as the ratio of convective mass transfer to the diffusion mass transport, and it is generally written in terms of the droplet Reynolds number, Re d , and Schmidt number, Sc, given by [62] Sh = 1 + (1 + Re d Sc) 1/3 f(Re d ). (2.43) The droplet Reynolds number is defined as the ratio of inertial forces to viscous forces, which is written as Re d = 2rρ g |u − v|/µ f . The Schmidt number is used to characterize
2.4. Single Droplet Modeling 31 the fluid flows in which there is simultaneous momentum and mass diffusion, and it is defined as the ratio between momentum diffusion and mass diffusion, written as Sc = µ f /(ρ g D f ). In Eq. (2.43), the function f(Re d ) depends upon the droplet Reynolds number, and in case of low Reynolds number, it may be calculated as defined by Abramzon and Sirignano [62], with f(Re d ) = 1 for Re d ≤ 1 and f(Re d ) = Re d 0.077 for Re d ≤ 400. In Eq. (2.41), B M is the Spalding mass transfer number, expressed in terms of the mass fraction of vaporized liquid as, B M = Y s − Y ∞ 1 − Y s . (2.44) Here Y s and Y ∞ are mass fractions of the water at the droplet surface and in the bulk of surrounding gas, respectively. Y s is computed from the vapor-liquid equilibrium through the vapor pressure of water, which is written as [148] Y s = M w M w + ¯M(¯p/p w − 1) . (2.45) The quantities M w and p w denote molar mass and vapor pressure of water while and ¯p represent molar mass and mean pressure of the surrounding gas, respectively. Although the initial temperatures of gas and the droplet are equal and are at room temperature, the droplet temperature is subject to change due to evaporation. Time evolution of droplet temperature for water spray is computed using the uniform temperature model [62], [ ] dT s mC pL dt = Q CpLf (T ∞ − T s ) L = ṁ − L V (T s ) , (2.46) B T where m is the droplet mass, Q L is the net heat transferred to the droplet per unit time, C pL and C pLf are the specific heat capacity of the liquid and in film, respectively, T s is the temperature at droplet surface, T ∞ is the temperature of the surrounding gas, and L V (T s ) is the temperature dependent latent heat of vaporization at T s . B T is the Spalding heat transfer number, which is calculated in terms of the mass transfer number using the relation [62] where the exponent φ is given by [62] B T = (1 + B M ) φ − 1, (2.47) φ = C pL ˜Sh 1 C pg Ñu Le . (2.48) Here C pg is the specific heat capacity of the gas, Le is the Lewis number, and ¯M Ñu is the modified Nusselt number, which accounts for convective droplet heating, and it is
Page 1: 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: 2.4. Single Droplet Modeling 29 col
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Page 55 and 56: 2.4. Single Droplet Modeling 35 gra
Page 57 and 58: 2.4. Single Droplet Modeling 37 thr
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Page 61 and 62: 2.4. Single Droplet Modeling 41 (a)
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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
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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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