INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

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50 3. Numerical Methods where U n ∈ {w n , w n ρ l r n , w n ρ l r n v n }, S n ∈ {a n , ρ l b n , ρ l c n }. To solve these equations, the choice of numerical scheme is important because this array of equations are strongly coupled. Different finite difference schemes with varying order of accuracy are tested. It has been shown [191] that Runge-Kutta 4 th order method can accurately solve the system of inhomogeneous equations represented by Eq. (3.19) and proven to be computationally efficient for DQMOM in one-dimensional physical space [191]. In the current study, the NDF is approximated by a three-node closure in DQMOM, which is proven to be accurate in previous studies [48, 49]. The three-node approximation of NDF implies that a total of nine coupled equations, which are generated by substituting n = {1, 2, 3}, in Eq. (3.19). These equations are solved to find the evolution of NDF, which is achieved by discretizing and estimating these equations with Runge-Kutta 4 th order method. At every spatial location within the geometry, the source terms are computed through the models proposed in Chapter 2. The flowchart shown in Fig. 3.1 outlines the step by step procedure of the computational code. The previous studies concerning DQMOM in spray flows, transport equations are never solved in two-dimensional configuration but only in one dimension. In this study, the DQMOM transport equations are solved in two-dimensional (axial and radial direction) geometrical configuration for water and PVP/water sprays in air, by implementing a finite difference numerical scheme. At each axial and radial location, the coupled steady state transport equations of DQMOM are solved and the source terms such as droplet heating, evaporation rate, total forces acting on droplet and droplet coalescence are computed from the weights and abscissas available from the initial values at first iteration and from the last computed value in the next iterations. The steady form of the DQMOM transport Eqs. (2.26) – (2.28) in two dimensions can be written as where ∂U n ∂x + ∂E n ∂z = S n, (3.21) U n ∈ {w n v n , w n ρ l r n v n , w n ρ l r n v n u n , w n ρ l r n v n v n }, E n ∈ {w n u n , w n ρ l r n u n , w n ρ l r n u n u n , w n ρ l r n u n v n }. (3.22) In Eq. (3.21), x is the axial direction, z is the radial direction, and the corresponding velocities are v and u, respectively. To keep the computational efficiency, ease of
3.2. Spray Modeling 51 application and numerical accuracy, a second order explicit finite difference scheme is applied to solve steady state form of Eqs. (2.26) – (2.28) [192], which are represented by Eq. (3.21). Thus the solution formula may be written as [193] U j+1 n,i = U j n,i − ∆x [ 1.5E j i ∆z − 2Ej i−1 + ] 0.5Ej i−2 + ∆xS j i , (3.23) where i and j are grid nodes in radial and axial directions, respectively. The above formulation is applied to an equidistant rectangular grid, where the size of each grid cell is 1.5 × 10 −3 m in radial direction and 1.0 × 10 −4 m in axial direction, resulting in a maximum of 80 × 1000 grid nodes. The initial data to start simulations in both the configurations, i.e, one and two-dimensional cases is generated from the experimental data provided by Dr. R. Wengeler, BASF Ludwigshafen (onedimensional water spray in nitrogen) and Prof. G. Brenn, TU Graz (two-dimensional water and PVP/water spray in air) using Wheeler algorithm (see Subsection 3.2.3). The experimental data closest to the nozzle exit is taken for generating the initial data and the procedure for calculating this initial data from experiment is explained in Chapter 4 along with brief description about the experimental setup. The boundary conditions in solving DQMOM include (1) if droplets hit the axis of symmetry, they are reflected, and (2) Neumann boundary is applied for the lateral sides of the computational domain and exit plane. The experimental data available at other cross sections away from the nozzle exit is used to validate the simulation results. The flowchart of the computational code is illustrated in Fig. 3.1. 3.2.3 Wheeler Algorithm The Wheeler algorithm developed by Sack and Donovan [136], requires 2N +1 moments to compute N weights (number density) and N abscissas (droplet radii or velocities). The moment set is represented as M = [M(0), M(1), ...M(2N + 1)] T . This algorithm is used to generate the initial data in DQMOM whereas in QMOM it is used to compute the unknown moments. The first step in Wheeler algorithm is to compute the coefficients π α based on these 2N + 1 moments of the distribution function n(ξ), given as π α+1 (ξ) = ξπ α (ξ). (3.24) The above recursive relation has the properties of π −1 (ξ) = 0 and π 0 (ξ) = 1. Here, α is a subset of number of moments 2N + 1, i.e., α ∈ 0, 1, 2..N − 1. From these coefficients π α (ξ), a symmetric tridiagonal matrix is computed through some intermediate quantities: ∫ σ α,β = n(ξ)π α (ξ)π β (ξ)dξ, (3.25)
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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
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 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: 3.2. Spray Modeling 49 ⎛ ⎞ ⎛
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
Page 119 and 120: 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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