MODELING CHAR OXIDATION AS A FUNCTION OF PRESSURE ...

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Results and Discussion Evaluation of the General Asymptotic Solution Values of the effectiveness factor predicted by the general asymptotic solution using the general moduli in Eqs. (4.2) and (4.3) were compared to the values of obtained by numerical solution. It was found that in spherical coordinates, the general asymptotic solution predicted the effectiveness factor with errors ranging from -17% to 0% on a relative basis (see Tables 4.1 and 4.2, and Figure 4.2). At the first order limit, the general asymptotic solution becomes an exact solution, and therefore the accuracy of the numerical solution was evaluated. The first column in Tables 4.1 and 4.2, corresponds to m = 1 and therefore represents the relative error between the numerical solution and the exact solution. It can be seen that at the first order limit, the error in the numerical solution for the m-th order model is less than 0.13%. The error in the numerical solution for the Langmuir rate equation is slightly higher (less than 1%). These small numerical errors likely arise due to the gridding scheme; even though the number of grid nodes used in the numerical model increases with M T (Eq. 4.15), the decrease in effectiveness factor means that the oxidizer penetration depth decreases, and hence only a small fraction of the total number of nodes have non-zero oxygen concentration. The fraction of nodes with non-zero oxygen concentration is approximately proportional to 1/M T when M T is greater than 5. Therefore, numerical errors as large as 1% were incurred for KC s = 0 and M T = 8. 32
Table 4.1. The Relative Error * (%) in the General Asymptotic Solution for m-th Order Rate Equations Using Eq. (4.2) MT m 1.00 ** 0.75 0.50 0.25 0.00 0.125 0.027 -0.109 -0.290 -0.543 -0.925 0.25 0.046 -0.441 -1.103 -2.058 -3.560 0.5 0.097 -1.316 -3.326 -6.462 -12.375 0.707 0.124 -1.818 -4.630 -9.156 -15.789 1 0.126 -1.963 -4.836 -8.546 -12.014 2 0.053 -1.174 -2.580 -4.177 -5.525 4 0.043 -0.548 -1.224 -1.941 -2.483 8 0.042 -0.260 -0.597 -0.896 -1.375 *Relative error = ( asymp - numerical)/ numerical ** asym = exact when m = 1.0 Table 4.2. The Relative Error* (%) in the General Asymptotic Solution for the Langmuir Rate Equation Using Eq. (4.3) M T 1/(1+KCs) 1.00 0.75 0.50 0.25 0.00 0.125 0.019 -0.162 -0.342 -0.583 -0.925 0.25 0.016 -0.588 -1.282 -2.188 -3.560 0.5 -0.013 -1.639 -3.672 -6.557 -12.375 0.707 -0.076 -2.162 -4.802 -8.618 -16.081 1 -0.215 -2.274 -4.756 -8.000 -12.392 2 -0.491 -1.584 -2.813 -4.277 -6.018 4 -0.679 -1.191 -1.774 -2.472 -3.156 8 -0.933 -1.186 -1.473 -1.821 -2.274 *Relative error = ( asymp - numerical)/ numerical ** asym = exact when 1/(1+KC s) = 1.0 33
Page 1 and 2: MODELING CHAR OXIDATION AS A FUNCTI
Page 3 and 4: BRIGHAM YOUNG UNIVERSITY As chair o
Page 5 and 6: CBK model uses: 1) an intrinsic Lan
Page 7 and 8: Table of Contents List of Figures..
Page 9: Appendices.........................
Page 12 and 13: Figure A.2. Mass releases of the Ko
Page 14 and 15: Table 7.6. Parameters Used in Model
Page 16 and 17: Ed activation energy of desorption,
Page 18 and 19: vc the volume of combustible materi
Page 21 and 22: Background 1. Introduction The rate
Page 23: the CBK model developed at Brown Un
Page 26 and 27: Zone III rate ∝ C og E obs → 0
Page 28 and 29: coal-general kinetic rate constants
Page 30 and 31: Boundary Layer Diffusion The molar
Page 32 and 33: = q obs q max The factor can be use
Page 34 and 35: where k 1 and K are two kinetic par
Page 36 and 37: particle can therefore be convenien
Page 38 and 39: This is the first time that the gen
Page 40 and 41: Data of Mathias Mathias (1996) perf
Page 42 and 43: urn with shrinking diameters, and t
Page 45 and 46: 3. Objectives and Approach The obje
Page 47 and 48: Introduction 4. Analytical Solution
Page 49 and 50: Task and Methodology Task One of th
Page 51: 2 [ (i +1) − (i − 1)] i b = −
Page 55 and 56: The resulting observations regardin
Page 57 and 58: correction. The values of functions
Page 59 and 60: Table 4.6. The Relative Error* (%)
Page 61 and 62: Table 4.8. The Relative Error* (%)
Page 63 and 64: general asymptotic solution. An arc
Page 65 and 66: 5. Theoretical Developments The int
Page 67 and 68: order of a reaction is usually dete
Page 69 and 70: nobs = 1 (KCs ) 2 2 1 [KCs − ln(1
Page 71 and 72: The observed reaction order in Zone
Page 73 and 74: Bulk Diffusion vs. Knudsen Diffusio
Page 75 and 76: where D K is in cm 2 /sec, r p is t
Page 77 and 78: where T p is in K, P is in atm. The
Page 79 and 80: Both of these assumptions are argua
Page 81 and 82: 2 r obs ′ − [kD Pog + k d + kD
Page 83 and 84: oxygen partial pressure (Suuberg et
Page 85 and 86: Farrauto and Batholomew (1997) prop
Page 87: assumes a homogeneous, non-interact
Page 90 and 91: Single-Film Char Oxidation Submodel
Page 92 and 93: where and An energy balance is used
Page 94 and 95: where is the empirical burning mode
Page 96 and 97: calculation uses a 7 × 7 × 7 matr
Page 98 and 99: HP-CBK Model Development The HP-CBK
Page 100 and 101: Effective Diffusivity The major obs
Page 102 and 103:
where r p1 and r p2 are the average
Page 104 and 105:
where r p1 is the macro-pore radius
Page 107 and 108:
7. Model Evaluation and Discussion
Page 109 and 110:
experiments are non-porous, the rat
Page 111 and 112:
and 2850 K). For consistency with t
Page 113 and 114:
The value of the roughness factor w
Page 115 and 116:
= S int S ext D e r p a 2 2M C M O2
Page 117 and 118:
Reactor Head Flow Straightener Reac
Page 119 and 120:
the large size of the particle, and
Page 121 and 122:
taking into account convection, rad
Page 123 and 124:
2.5x10 -4 2 /sec) 2.0 1.5 Rate (g/c
Page 125 and 126:
Table 7.5. The Experimental Conditi
Page 127 and 128:
The burnout and particle velocity d
Page 129 and 130:
The HP-CBK was used to predict the
Page 131 and 132:
TGA and FFB Data-This Study The rea
Page 133 and 134:
This equation can be derived as fol
Page 135 and 136:
q = A 1p e − E 1 p / RT P os 1 +
Page 137 and 138:
m obs = 0 at high temperatures) and
Page 139 and 140:
Currently the correlations between
Page 141 and 142:
8. Summary and Conclusions The obje
Page 143 and 144:
0.5 due to the contribution from th
Page 145 and 146:
Langmuir rate equation, the reactio
Page 147 and 148:
II, in agreement with many observat
Page 149 and 150:
9. Recommendations The predictive c
Page 151 and 152:
References Ahmed, S., M. H. Back an
Page 153 and 154:
Essenhigh, R. H., D. Fortsch and H.
Page 155 and 156:
Mehta, B. N. and R. Aris , “Commu
Page 157 and 158:
Szekely, J. and M. Propster, "A Str
Page 159 and 160:
Appendices 139
Page 161 and 162:
Introduction Appendix A: Experiment
Page 163 and 164:
detaching the flame from the burner
Page 165 and 166:
To study the effects of steam, CO w
Page 167 and 168:
times at heights of 1, 2, 4, and 6
Page 169 and 170:
analysis. The char reactivities (in
Page 171 and 172:
Table A.5. Moisture, Ash and ICP Ma
Page 173 and 174:
Table A.9. Elemental Analyses of Fo
Page 175 and 176:
temperature profile of the post-fla
Page 177 and 178:
Apparent densities 1.00 0.75 0.50 0
Page 179 and 180:
This observation is somewhat surpri
Page 181 and 182:
It is interesting to compare Figure
Page 183 and 184:
The N 2 BET surfacea areas and H/C
Page 185 and 186:
collected in the #4 reactor conditi
Page 187 and 188:
Rate (gC /g C remaining /sec) 1.6x1
Page 189 and 190:
close to zero, the accumulated erro
Page 191:
Appendix B: Errors and Standard Dev
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MODELING CHAR OXIDATION AS A FUNCTION OF PRESSURE ...

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