INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

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76 4. Results and Discussion experimental data are somewhat spread at 0.12 m from the nozzle exit. At 0.16 m, the slopes of the numerical results deviate from the experimental data, particularly in large size droplets region where the effective cross sectional area shows strong fluctuations in experiment as shown in Fig. 4.14. This implies that the post-processing of experimental data plays an important role in the corrections of number density and thereby the droplet properties [207]. Comparing the velocity profiles at the two different cross sections, it is seen that the velocity decreases as droplets move away from the nozzle. This is because the droplets are strongly decelerated by the dynamic interaction with the surrounding gas. The gas around the spray stagnates and is driven into motion only due to the spray entrainment. The gas motion driven by the spray arises at the expense that the droplet loses momentum. The droplet properties are predicted quite well by the present simulations, which confirms their applicability for spray flows. There are some deviations between simulation and experimental results, which are attributable to the post-processing of the experimental data as mentioned before. In case of DDM, neglecting droplet–droplet interactions may need reconsideration. For DQMOM, the improved numerical scheme and the simultaneous solution of the gas phase equations may improve the simulation results. Based on these simulation results and comparison with the experiment, it can be concluded that the DQMOM is a robust method, which can predict the spray flows accurately. This led to the implementation of DQMOM to study bi-component evaporating spray, i.e., PVP/water spray flow in two dimensions. In order to perform simulations of PVP/water spray flows, the predictability and efficiency of developed bicomponent droplet evaporation and solid layer formation model (see Subsection 2.4.1.2) needs to be verified under different drying conditions. The next section presents the numerical simulation of single bi-component droplet evaporation and solid layer development, and comparison of simulation results with experiment. 4.3 Single Bi-component Droplet Evaporation and Solid Layer Formation The model presented in Subsection 2.4.1.2 to predict the evaporation and solid layer formation for PVP/water droplet and mannitol/water droplet is simulated with different conditions such as initial solute mass fraction, gas temperature and velocity, relative humidity, initial droplet size etc. In the next subsections, the vapor-liquid equilibrium calculation followed by non-ideality effect caused by the solute (PVP or mannitol) presence on the droplet heating and evaporation rate is explained. Finally, the single droplet evaporation and solid layer development results are presented.
4.3. Single Bi-component Droplet Evaporation and Solid Layer Formation 77 4.3.1 Vapor-Liquid Equilibrium The vapor-liquid equilibrium for the evaporating component i is needed for the calculation of Spalding’s mass transfer number, B M,1 , cf. Eq. (2.54), for both PVP/water and mannitol/water droplet evaporation and solid layer development cases, in which the mass fraction, Y 1,s , of the evaporating component appears; this mass fraction is calculated through the mole fraction, X i , i = 1, of the evaporating component water, in terms of the activity coefficient X i = p vap,i p m γ i X L,i , (4.4) where p vap,1 is the vapor pressure of pure water and p m is the total mixture pressure, which is equal to the ambient gas pressure, and in the present study it equals the atmospheric pressure. Here, X L,i is the mole fraction of evaporating component i in liquid phase, and γ i is the activity coefficient of evaporating component i, which is calculated through equation given as, γ i = a wY L,i X L,i . (4.5) Here a w is the water activity, Y L,i with i = 1, is the water mass fraction within the droplet. The calculation of water activity coefficient is described in next subsection. 4.3.2 Non-ideal Liquid Mixture The presence of polymer or mannitol with water leads to non-ideal liquid behavior, which must be accounted for in calculating the mole fraction of water vapor at the droplet surface. In this work, the liquid mixture is treated as non-ideal by determining the influence of individual components on each other through their activity coefficients. The universal functional activity coefficient (UNIFAC) method is the accurate and most extensively used procedure [210], which estimates the activity coefficient as a sum of combinatorial and residual terms. This method, however, cannot be applied for polymer solutions as they have significant difference in accessible volume for a molecule in the solution [211]. The work of Oishi and Prausnitz [211] extended UNIFAC method to account for such differences in accessible volume by introducing a free-volume term, which enabled the UNIFAC approach to be applied to polymer solution systems. However, it is proven that their model fails for aqueous polymer systems because of the inadequacy of its free-volume term [212]. In the current study, the activity coefficient of water in PVP/water solution is computed using the UNIFAC-van der Waals-Free Volume method known as UNIFAC-vdW-FV method [212], which accounts for the free-volume
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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
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2.2. Euler - Lagrangian Approach 17
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2.2. Euler - Lagrangian Approach 19
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2.3. Euler - Euler Approach 21 quan
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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 and 88: 4.2. Two-dimensional Evaporating Wa
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Page 99 and 100: 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
Page 138 and 139: BIBLIOGRAPHY vii [72] G. M. Faeth:
Page 140 and 141: BIBLIOGRAPHY ix [95] C. Cercignani,
Page 142 and 143: BIBLIOGRAPHY xi [119] R. O. Fox: Hi
Page 144 and 145: 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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