Diffusion Reaction Interaction for a Pair of Spheres - ETD ...

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Figure 3.12: Constant concentration c c0 contour curves for two diffusion-controlled sinks. Sphere 2 the smaller sink ( 2 1 = a ) is shown at the center. Only the lower portion of he larger sink 1 ( a 10 ) is shown −1 at the top. The center-to-center separation distance d( a1 + a2 ) = 1. 01 is taken near closest approach contact for maximum recovery of R1 from the minimum. 86 1 =
CHAPTER 4 EXACT SOLUTION FOR THE CONSTANT CONCENTRATION 4.1 Introduction SOURCE – REACTIVE SINK PROBLEM In Chapter 2 the sink source mutualism like interaction was considered where the source had a constant surface generation rate (zeroth order kinetics). The sink in the aforementioned problem was modeled as a first order interfacial reactor. There are many models where the source medium interface exhibits a different behavior. In some catalytic conversion examples the source may require first order or Langmuir Hinshelwood rate constants (Fogler, 1999) while dissolution processes may involve a constant concentration at the source –medium interface (Forsten and Lauffenburger, 1992 and Ham, 1958 and 1959). Primary examples include: crystal growth and coarsening (Voorhees and Schaefer, 1987), drop coalescence (Lifshitz and Pitaevskii, 1981), electric current and heat flow (Stoy, 1989b) and constant oxygen concentration at a source for delivery to microorganisms in the remediation of contaminated soils (Charackliss and Marshall, 1990). 87
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DIFFUSION INTERACTIONS FOR A PAIR O
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Nyrée McDonald bispherical coordin
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TABLE OF CONTENTS FIGURES………
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FIGURES 1.1 Two spheres of radius a
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3.3 Dimensionless consumption rate
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3.12 Constant concentration 0 c c c
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5.5 Reaction probability Pint for a
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B.1 The bispherical coordinates (μ
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a Radius of sphere 1 1 a Radius of
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K Matrix elements for the probabili
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η Bispherical coordinate θ1 θ2 S
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CHAPTER 1 HISTORICAL REVIEW OF DIFF
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compounds (Siebel and Charcklis, 19
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the end, Tsao (2001) claimed that t
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1990), permeable (Torquato and Avel
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4 that for site volumes fractions 1
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internal reaction in sphere 1 and c
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directly to the concentration and t
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CHAPTER 2 DIFFUSION REACTION FOR A
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2.2 Mutualism Interaction Between a
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P 2πa Dc 2 π 1 0 = ⎡∂u ⎤ si
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P ( cosΘ1) n ( n+ 1) r1 1 = n d 1
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2.4 Mutualism-like Problem: Analyti
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h 10 ∑ m 1 ∞ = 1+ K = 1− K 0m
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where and ( 2) ( 2) h = Q + h K , n
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They can be calculated exactly for
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Similarly, the coefficient h can be
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coefficients f1n. Since h1n was act
Page 57 and 58: G ( 1) n ( 0) ( 0) G0 K = Gn + , (2
Page 59 and 60: probability P (2.55), and the conce
Page 61 and 62: ate constant λ 50 . The figures 2.
Page 63 and 64: Figure 2.2 - 2.4 shows the monotoni
Page 65 and 66: Θ1 d Figure 2.1: Two spheres of ra
Page 67 and 68: Figure 2.3: Reaction probability P
Page 69 and 70: Figure 2.5: Constant concentration
Page 71 and 72: CHAPTER 3 COMPETITIVE INTERACTION B
Page 73 and 74: a2, respectively. Any point externa
Page 75 and 76: The overall reaction rate at sphere
Page 77 and 78: where Λ kn ⎛ ak = ⎜ ⎝ d n
Page 79 and 80: ∑( ) ∞ − j ⎛ j + n⎞⎛ j
Page 81 and 82: and ( ) 1 − 1− L ( i+ 1) () i (
Page 83 and 84: i→∞ ( n+ i) ( ∞) v1 n = lim M
Page 85 and 86: 1953). The reaction rate for this c
Page 87 and 88: a1 sinh μ1 h μ = , (3.47) cosh μ
Page 89 and 90: ispherical coordinate system holds
Page 91 and 92: eaction rate remains virtually unch
Page 93 and 94: on the x = 0 line with the larger s
Page 95 and 96: contact (d=a1+a2) to far apart (d>>
Page 97 and 98: Figure 3.1: Dimensionless consumpti
Page 107: Figure 3.11: Constant concentration
Page 111 and 112: for all points external to the sink
Page 113 and 114: R 1 π π ⎡ ⎤ , ⎢⎣ ∂r1
Page 115 and 116: and −1⎛ f ⎞ μ = ⎜ ⎟ 1 si
Page 117 and 118: and the small number of infinite su
Page 119 and 120: 4.4 Results and Discussion Bispheri
Page 121 and 122: 4.5 Summary and Conclusions Exact a
Page 125 and 126: CHAPTER 5 DIFFUSION FROM A SPHERICA
Page 127 and 128: more concise expression for the int
Page 129 and 130: across the interface where the para
Page 131 and 132: This Pint is the ratio of the amoun
Page 133 and 134: integral is reduced to 2 π times t
Page 135 and 136: ( cosθ 1) Pn 1 m ⎛ m + n⎞⎛ r
Page 137 and 138: ′ i particularly for n = 0, n ′
Page 139 and 140: where int D D = ε , Γ ( φ ) = I
Page 141 and 142: T ( i+ 1) ( i) n = T n ( ) ( i) ( i
Page 143 and 144: where ε is the ratio of external t
Page 145 and 146: equation (5.14) and u the external
Page 147 and 148: Figure (5.7) where ( γ = 0. 1 and
Page 149 and 150: approach [ d a + a ) = 1], small Th
Page 151 and 152: Figure 5.2: Reaction probability Pi
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Figure 5.10: Reaction probability P
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APPENDIX A ANALYTICAL FORMS OF Fnm
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APPENDIX B TWO SPHERES IN BISPHERIC
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with center at the origin, and the
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c ( r ) = c (r far from either sphe
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B.3 Reaction Rate The reaction rate
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where c is the concentration of the
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and dP d cos ∑( ) ≥ − − m 2
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The rate is scaled by the Smoluchow
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Sphere 2 μ =−μ2 y u φ x η= η
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[11] Clesceri, L., A. E. Greenberg,
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[37] Lifshitz, E. M. and L.P. Pitae
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[60] Regalbuto, M. C., Strieder, W.
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[84] Torquato, S., “Diffusion and
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