INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

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34 2. Mathematical Modeling Fig. 2.5: Schematic diagram of stages in single droplet evaporation and drying. model, achieving fairly good agreement. The droplet generators used in these studies produce a chain of closely spaced droplets, which leads to droplet–droplet interactions in processes that are limited by gas phase transport processes. The aim of the present work is to develop a mathematical model, which can be applied to predict the evaporation and drying characteristics of droplets of the polymer PVP dissolved in water and mannitol dissolved in water solution. Prerequisites of the method are, (1) accounting for the solid layer resistance in mass evaporation rate and energy calculation, and (2) treatment of the liquid mixture as non-ideal by computing the activity coefficient of the evaporating component. The problem under consideration is the evaporation and drying of an isolated single spherical droplet consisting of a binary mixture of a liquid and a dissolved solid material with low or zero vapor pressure. During the evaporation and drying of the bi-component droplet, the droplet undergoes four stages as explained by Nesic and Vodnik [151], which are depicted in Fig. 2.5. In the initial stage, the droplet temperature quickly rises to an equilibrium temperature, which is most often near to the wet bulb temperature for surrounding gas and humidity, with some solvent evaporation. In the second stage, the droplet starts to shrink as solvent evaporates causing the solute mass fraction to increase at the droplet surface; this leads to a slight raise in the droplet temperature (see Fig. 2.5). The increase in solute mass fraction at the droplet surface hinders further evaporation as the vapor pressure of the solvent at the surface drops. The third stage of drying starts when the solute mass fraction at the surface raises to a threshold value, which most often is equal to the saturation solubility of the solute in the solvent, whereupon the crust formation starts for salts, sugar and colloidal material. In the case of polymers, molecular entanglement and
2.4. Single Droplet Modeling 35 gradual increase in concentration lead to solid layer formation at the droplet surface. In the latter case, the solid layer thickens and develops into the droplet interior as shown in Fig. 2.5, and a rapid fall in evaporation rate is observed. In this period, the heat penetrated into the liquid is used for heating the droplet, which causes the droplet temperature to rise rapidly. Further drying behavior of droplet depends on the vapor diffusivity through the solid layer. In the final stage of drying, boiling followed by particle drying, eventually leading to dried product formation, takes place. The different assumptions in developing this mathematical model include the following: 1. The droplet remains spherical in shape throughout the evaporation with spherical symmetry. 2. Solubility of gas in liquid is negligible. 3. Gas phase is in a quasi-steady state. 4. No influence of chemical reactions occurs within and outside the droplet. 5. No heat transfer due to radiation. 6. No mass diffusion by temperature and pressure gradients. 7. No change in droplet radius once the solid layer formation starts. 8. Internal circulation of water and capillary effects are negligible. The problem of evaporation and drying of a single droplet can be well defined using the species mass diffusion and heat conduction equations in spherical coordinates. The diffusion equation for the substance i in the droplet, formulated in terms of mass fraction Y i , reads ∂Y i ∂t = D [ ( 12 ∂ r 2 ∂Y )] i , (2.51) r 2 ∂r ∂r where D 12 is the binary diffusion coefficient in the liquid, r is the radial coordinate within the droplet radius, and t stands for time. In this equation i = 1 denotes the solvent (water) and i = 2 denotes solute (PVP or mannitol). Initially, the droplet is a homogenous mixture, Y i = Y i0 at t = 0 s. At the droplet center, r = 0 m, the regularity condition must be satisfied at any time, ∂Y i /∂r = 0. The boundary condition at the droplet surface must account for the change in droplet size, ∂Y i −D 12 ∂r − Y ∂R i ∂t = ṁi Aρ l (2.52) at r = R(t). Here ṁ i is the mass evaporation rate of substance i across the droplet surface, R(t) and A(t) are time dependent droplet radius and surface area, respectively,
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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: 2.4. Single Droplet Modeling 33 pol
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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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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