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

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

Diffusion Reaction Interaction for a Pair of Spheres - ETD

DIFFUSION INTERACTIONS FOR A PAIR OF REACTIVE SPHERES A Dissertation Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Nyrée V. McDonald, B.S., M.S. William Strieder, Director Graduate Program in Chemical and Biomolecular Engineering Notre Dame, Indiana November 2005

  • Page 2 and 3: DIFFUSION INTERACTIONS FOR A PAIR O
  • Page 4 and 5: DEDICATION This is for my daughter,
  • Page 6 and 7: 3.2 Competitive Interaction: Two Si
  • Page 8 and 9: 2.5 Constant concentration contour
  • Page 10 and 11: 3.7 Dimensionless consumption rate
  • Page 12 and 13: 5.2 Reaction probability Pint for a
  • Page 14 and 15: 5.8 Reaction probability Pint for a
  • Page 16 and 17: TABLES 5.1 The results show the per
  • Page 18 and 19: f Equation coefficient of sphere 2
  • Page 20 and 21: s Modified Bessel function of the f
  • Page 22 and 23: ACKNOWLEDGMENTS Special thanks to t
  • Page 24 and 25: Marshall, 1990; Converti et al., 19
  • Page 26 and 27: problems focused on diffusion to tw
  • Page 28 and 29: the size and reactivity of the inte
  • Page 30 and 31: In the past most efforts to underst
  • Page 32 and 33: Hassan, 1985; Teixeira et al., 1994
  • Page 34 and 35: from the set of linear equations. T
  • Page 36 and 37: d a1 a2 Figure 1.1: Two spheres of
  • Page 38 and 39: cells have to be before they intera
  • Page 40 and 41: The intermediate product is consume
  • Page 42 and 43: and the dimensionless center-to-cen
  • Page 44 and 45: condition of the Legendre polynomia
  • Page 46 and 47: with the substitution of (2.20) int
  • Page 48 and 49: and ( 0) K = K nm nm . (2.30) Then
  • Page 50 and 51: () () ∑ ∞ h1 n i i = Qn + K nmh
  • Page 52 and 53:

    It is important to recognize that e

  • Page 54 and 55:

    So, by eliminating all for m > 0 an

  • Page 56 and 57:

    ∑ ∞ = 1n n n= 0 G h u , 1 d1 a1

  • Page 58 and 59:

    G ( 3) n ( 2) ( 2) G2 K = Gn + , (2

  • Page 60 and 61:

    4πa Dc = 1 m R m . (2.71) ( 1+ λ1

  • Page 62 and 63:

    λ 1 the slope of the curve is alwa

  • Page 64 and 65:

    intermediate product (growth factor

  • Page 66 and 67:

    Figure 2.2: Reaction probability P

  • Page 68 and 69:

    Figure 2.4: Reaction probability P

  • Page 70 and 71:

    Figure 2.6: Constant concentration

  • Page 72 and 73:

    Finally, the bispherical coordinate

  • Page 74 and 75:

    Note, boundary conditions (3.2), (3

  • Page 76 and 77:

    where Pn is the Legendre polynomial

  • Page 78 and 79:

    1 ∫ − 1 is P ( z) P ( z) n m 2

  • Page 80 and 81:

    where and Λ are given by equation

  • Page 82 and 83:

    If equation (3.25) is written for n

  • Page 84 and 85:

    3.4.2 Diffusion Limited Concentrati

  • Page 86 and 87:

    where = − − ∑ ∞ κ cosh μ

  • Page 88 and 89:

    3.5 Results And Discussion Competit

  • Page 90 and 91:

    of the size ratio / ] a = γ = 0.1,

  • Page 92 and 93:

    eactivity 1 λ and radius a there a

  • Page 94 and 95:

    less and less of the larger sink 1

  • Page 96 and 97:

    arising from possible size and reac

  • Page 98 and 99:

    Figure 3.2: Dimensionless consumpti

  • Page 100 and 101:

    Figure 3.4: Dimensionless consumpti

  • Page 102 and 103:

    Figure 3.6: Dimensionless consumpti

  • Page 104 and 105:

    Figure 3.8: Dimensionless consumpti

  • Page 106 and 107:

    Figure 3.10: Dimensionless consumpt

  • Page 108 and 109:

    Figure 3.12: Constant concentration

  • Page 110 and 111:

    In this chapter the exact steady st

  • Page 112 and 113:

    subtracting the last scenario from

  • Page 114 and 115:

    -∞

  • Page 116 and 117:

    such that the concentration in the

  • Page 118 and 119:

    2 sinh μ1 d(cosθ ) = d(cosη) . (

  • Page 120 and 121:

    μ → ∞ and μ → ∞ and all c

  • Page 122 and 123:

    Figure 4.1: Dimensionless consumpti

  • Page 124 and 125:

    Figure 4.3: Dimensionless consumpti

  • Page 126 and 127:

    species. Therefore, commensalism is

  • Page 128 and 129:

    2 D c = 0 (5.1) ∇ ext where D is

  • Page 130 and 131:

    the dimensionless center-to-center

  • Page 132 and 133:

    of the first kind of half-integer o

  • Page 134 and 135:

    5.4 External Concentration The dime

  • Page 136 and 137:

    The final set of coefficients b , a

  • Page 138 and 139:

    5.5 Internal Reaction Probability:

  • Page 140 and 141:

    P β 0 int = − g10 . (5.35) d1 No

  • Page 142 and 143:

    is accomplished by setting the effe

  • Page 144 and 145:

    and ( ) 0 Q = 1. (2.43) n They are

  • Page 146 and 147:

    epresents P and a solid line repres

  • Page 148 and 149:

    difference is − 4. 48% and forε

  • Page 150 and 151:

    Figure 5.1: Reaction probability Pi

  • Page 152 and 153:

    Figure 5.3: Reaction probability Pi

  • Page 154 and 155:

    Figure 5.5: Reaction probability Pi

  • Page 156 and 157:

    Figure 5.7: Reaction probability Pi

  • Page 158 and 159:

    Figure 5.9: Reaction probability Pi

  • Page 160 and 161:

    TABLE 5.1 PERCENT ERROR FOR THE EXA

  • Page 162 and 163:

    x d ⎧1 d = ⎨ m! dx ⎩ x dx [

  • Page 164 and 165:

    2 ⎡ ⎤ 2 1 ∂ ⎛ ∂ψ ⎞ 1

  • Page 166 and 167:

    ⎡ 1 ⎤ m cos cosh μ − cosη e

  • Page 168 and 169:

    c0 − ci 0 = c 0 2 [ − ( n + 1 2

  • Page 170 and 171:

    and Also, sinη sinh μ −1 1− c

  • Page 172 and 173:

    ∂c 1 = − sinh μ1 ∂μ 2 −

  • Page 174 and 175:

    1 ∫ − 1 2δ nm Pn ( z) Pm ( z)

  • Page 176 and 177:

    −μ μ=−μ2 μ=-∞ z=-f f sinh

  • Page 178 and 179:

    REFERENCES [1] Abramowitz, M. and S

  • Page 180 and 181:

    [23] Fradkov, V. E., Glickman, M. E

  • Page 182 and 183:

    [49] Morse, P.M. and H. Feshbach, M

  • Page 184 and 185:

    [73] Smoluchowski, M. “Zusammenfa

  • Page 186:

    [97] Weisz, P. B., “Diffusion and

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