MODELING CHAR OXIDATION AS A FUNCTION OF PRESSURE ...
MODELING CHAR OXIDATION AS A FUNCTION OF PRESSURE ... MODELING CHAR OXIDATION AS A FUNCTION OF PRESSURE ...
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
Results and Discussion<br />
Evaluation of the General Asymptotic Solution<br />
Values of the effectiveness factor predicted by the general asymptotic solution<br />
using the general moduli in Eqs. (4.2) and (4.3) were compared to the values of obtained<br />
by numerical solution. It was found that in spherical coordinates, the general asymptotic<br />
solution predicted the effectiveness factor with errors ranging from -17% to 0% on a<br />
relative basis (see Tables 4.1 and 4.2, and Figure 4.2).<br />
At the first order limit, the general asymptotic solution becomes an exact solution,<br />
and therefore the accuracy of the numerical solution was evaluated. The first column in<br />
Tables 4.1 and 4.2, corresponds to m = 1 and therefore represents the relative error<br />
between the numerical solution and the exact solution. It can be seen that at the first<br />
order limit, the error in the numerical solution for the m-th order model is less than 0.13%.<br />
The error in the numerical solution for the Langmuir rate equation is slightly higher (less<br />
than 1%). These small numerical errors likely arise due to the gridding scheme; even<br />
though the number of grid nodes used in the numerical model increases with M T (Eq.<br />
4.15), the decrease in effectiveness factor means that the oxidizer penetration depth<br />
decreases, and hence only a small fraction of the total number of nodes have non-zero<br />
oxygen concentration. The fraction of nodes with non-zero oxygen concentration is<br />
approximately proportional to 1/M T when M T is greater than 5. Therefore, numerical<br />
errors as large as 1% were incurred for KC s = 0 and M T = 8.<br />
32
Change language