INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

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44 2. Mathematical Modeling where K is given as [186] K = 2 ρ l |v 1 − v 2 |r2 2 . (2.76) 9 µ g r 1 For DQMOM, substitution of all these sub-models defined in Subsection 2.4 into the Eq. (2.29) results in a linear system, which in turn is substituted into Eq. (2.30) to yield in a linear system of equations. The solution of this linear system gives the various source terms, i.e., a n , b n and c n that appear in DQMOM transport Eqs. (2.26), (2.27) and (2.28), which then constitutes a closed system. The numerical solution procedure to solve these equations with initial and boundary conditions is explained in Chapter 3. Droplet–droplet interactions are currently neglected in DDM because of the computational complexity involved if Lagrangian models are used. In summary, the new implementations of the present study include implementation of an advanced droplet evaporation model for water sprays in DQMOM, a new mathematical model development for the polymer or sugar dissolved in water droplets evaporation and solid layer formation at the droplet surface, and improvement of the evaporative flux calculation with maximum entropy formulation. Extension of DQMOM to simulate two-dimensional system, and implementation of developed bi-component evaporation model in DQMOM.
3. Numerical Methods In the numerical simulation, the governing equations are discretized and solved by computer programs where appropriate numerical algorithms are required. An ideal numerical algorithm should • be linearly stable for all cases of interest; • ensure the positivity property when appropriate; • be reasonably accurate; • be computationally efficient. There are several numerical methods available for the fluid mechanics. The methods ranging from the most discrete (or particulate) in nature to the most continuous (or global) include: • particle methods • characteristic methods • Lagrangian finite difference/finite volume method • Eulerian finite difference/finite volume method • finite element methods • spectral methods Each method has advantages and disadvantages, consequently has the preferable applications. Usually, it is difficult or inefficient for a stand-alone method to simulate a complex system. Hybrid method, which is like a bootstrapping process, combines the advantages of the multiple methods and minimizes their disadvantages. The disadvantage of hybrid method is that the consistency problem is more serious. Special strategies are needed to keep consistency between the multiple methods. In this chapter the numerical methods employed to solve the single droplet evaporation and drying equations, QMOM and DQMOM transport equations are explained. In this work, the DDM computations performed by Humza [68] are used to validate the DQMOM results for water spray in air in two-dimensional, axisymmetric configuration. Hence, the numerical details of the DDM simulations can be referred to Humza [68].
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INAUGURAL-DISSERTATION zur Erlangun
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Dreams can never become a reality w
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II ial direction). Earlier studies
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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 and 58: 2.4. Single Droplet Modeling 37 thr
Page 59 and 60: 2.4. Single Droplet Modeling 39 whe
Page 61 and 62: 2.4. Single Droplet Modeling 41 (a)
Page 63: 2.4. Single Droplet Modeling 43 not
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
Page 113 and 114: 4.4. Two-dimensional Evaporating PV
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4.4. Two-dimensional Evaporating PV
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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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