INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

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52 3. Numerical Methods Initialization (Generate weights and abscissas from the moments through Wheeler algorithm) Grid generation Compute droplet evaporation, Eq. (2.55) Compute droplet heating, Eq. (2.59) Compute drag force, Eq. (2.63) Compute coalescence, Eq. (2.69) Solve source terms of spray, Eq. (2.30) Update weights, abscissas, Eqs. (2.26) – (2.28) No Is the final measurement position reached? Stop Yes Fig. 3.1: Flowchart of the DQMOM computational code.
3.3. Numerical Performance 53 where β ∈ α, α + 1, ...2N − α − 1. These quantities, σ α,β , are calculated by initializing and a 0 = M(1)/M(0), b 0 = 0. The recurrence relation is σ −1,α = 0, (3.26) σ 0,α = M(α), (3.27) σ α,β = σ α−1,β+1 − a α−2 σ α−1,β − b β−1 σ α−2,β , (3.28) where the tridiagonal matrix components are given as a α = σ α,α+1 − σ α−1,α , (3.29) σ α,α σ α−1,α−1 σ α,α b α = . (3.30) σ α−1,α−1 Here, the values of a α are the diagonal elements and b α are the upper and lower diagonal elements of the symmetric tridiagonal matrix. The eigenvalues of this matrix are the abscissas (droplet radii, velocities) where as the corresponding eigenvectors are the weights (number densities). More details about <strong>der</strong>ivation of this algorithm is given by Gautschi [194], and example calculations are given by Marchisio and Fox [71]. 3.3 Numerical Performance For the DDM computations, which are carried out Humza [68], a hybrid finite volume method based on the SIMPLER (Semi-Implicit Method for Pressure-Linked Equations - Revised) algorithm [68, 195] is used to solve the mean conservation equation of the gas flow, and a Lagrangian stochastic droplet parcel method is used for the spray flow. The initial and boundary conditions are generated from the experimental data. A non-equidistant rectangular numerical grid is used, which is finer in the region near the nozzle exit with a total of 78 × 101 grid nodes. The numerical time step for the governing gas phase equations is controlled by applying the CFL condition [188]. The solution algorithm and numerical details of the DDM calculation are given by Humza [68]. The DQMOM simulations are carried out on a PC with two Intel dual core 2.2 GHz processors having 8 GB RAM. The DDM is simulated on a PC having an AMD quad Opteron 1.8 GHz processor with 64 GB RAM [68]. The latter PC had several jobs running simultaneously, so that the available RAM on both the PCs is about identical. All simulations are run on a single processor. The computations for DQMOM and DDM take about one hour and three days, respectively. Thus, the DQMOM computations show a much better performance with respect to the computational cost.
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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
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 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: 3.2. Spray Modeling 51 application
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
Page 119 and 120: 5. Conclusions and Future Work The
Page 121 and 122: 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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