INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

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68 4. Results and Discussion 200 µm as shown in Fig. 4.14 is invalid as the effective cross-sectional area should increase with droplet size [207]. The effective cross-section area is therefore calculated using a linear trend line from a threshold diameter. In the first step, the linear trend line is calculated using a linear regression scheme based on the data in the droplet size classes up to 60% of the maximum droplet size. In the second step, for all droplet size classes larger than 40% of the maximum droplet size class for this experimental position, the values of the effective cross sectional area are obtained as values of the linear trend line. Therefore, there is an overlap of the size class ranges used for computing the trend line and those whose probe volume cross-section areas are calculated using the trend line. Once the effective cross-sectional area probe volume is corrected, the number density is corrected correspondingly. Then, the moment sets of droplet size and velocities are computed, which in turn are used to calculate the initial weights (number densities), radii and velocities using the Wheeler algorithm [136]. In the present study, the spray distribution is approximated by a three-node closure, which is proven to be accurate in previous studies [48, 49, 191, 202]. The three-node approximation of NDF implies that the required number of moments is 12 (3 each: weights, droplet radii, axial velocities, radial velocities). The same procedure is followed at every radial position for the crosssection of 0.08 m. Figure 4.15 shows the experimental distribution of droplets and DQMOM approximation at 0.066 m from the center of the spray for 80 kg/h water flow rate. The problem of negative moments is handled by employing the adaptive Wheeler algorithm [208]. Number density [ (µm) 1 ] 0.05 0.04 0.03 0.02 0.01 Experiment DQMOM 0 0 50 100 150 Droplet radius [ µm ] Fig. 4.15: Experimental and DQMOM approximation of droplet number density for a water spray.
4.2. Two-dimensional Evaporating Water Spray in Air 69 The DDM simulation are carried out only for water spray in two-dimensional configuration by Humza [68]. In his work, the same experimental data is used to generate a system of parcels for DDM, where the properties of the k th parcel are denoted by (x k , r k , u k , v k , m k ) for the present two-dimensional configuration. The liquid mass of k th parcel is computed assuming the spherical symmetry of the droplets, i.e., m k = N∑ i=1 4 3 πρ lr 3 i , (4.3) where N refers to the number of droplets in the parcel. The number of parcels for the inflow rate of 80 kg/h is 3,704 and for 120 kg/h, it is 3,464. A non-equidistant rectangular grid with 7,878 grid points (78 in radial and 101 in axial direction, respectively) is used [68]. 4.2.3 Results and Discussion At first the implementation of maximum entropy (ME) method for the calculation of evaporative flux is studied. Fig. 4.16 shows the computed NDF at radial distance of 64.5 mm from the center and 0.08 m axial distance from nozzle for 80 kg/h liquid flow rate using ME method and its comparison with experiment, where a good agreement between the ME approximated NDF and experiment can be found. The evaporative flux computed using the weight ratio constraints, which are defined by Eqs. (2.33)– (2.34), for this position is found to be 0.39, where as with ME method it is 0.022 and 0.05 0.04 Flow rate: 80 kg/h Axial position: 80 mm Radial position: 64.5 mm ME reconstruction Experiment NDF [(µm) 1 ] 0.03 0.02 0.01 0 0 25 50 75 100 125 150 175 Droplet radius [µm] Fig. 4.16: Experimental and reconstructed NDF of 80 kg/h water spray at 64.5 mm from the center of the spray axis.
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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-
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Contents Abstract . . . . . . . . .
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List of Tables 4.1 Initial weights
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List of Figures 1.1 A schematic dia
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List of Figures XIII 4.27 Effect of
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1. Introduction Drying has found ap
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3 Fig. 1.2: Chemical structure of p
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5 via inhalation aerosols. Dry powd
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7 equations are written in terms of
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9 for through appropriate sub-model
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2. Mathematical Modeling Spray cons
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2.1. State of the Art 13 the govern
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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 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: 4.2. Two-dimensional Evaporating Wa
Page 91 and 92: 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
Page 119 and 120: 5. Conclusions and Future Work The
Page 121 and 122: 101 proved UNIFAC-vdW-FV method for
Page 123: Appendix
Page 126 and 127: 106 A. Nomenclature Symbol Unit Des
Page 128 and 129: 108 A. Nomenclature S n S g S l Vec
Page 130 and 131: 110 A. Nomenclature List of Abbrevi
Page 132 and 133: Bibliography [1] K. Masters: Spray
Page 134 and 135: BIBLIOGRAPHY iii [24] K. D. Squires
Page 136 and 137: 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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