MODELING CHAR OXIDATION AS A FUNCTION OF PRESSURE ...
MODELING CHAR OXIDATION AS A FUNCTION OF PRESSURE ... MODELING CHAR OXIDATION AS A FUNCTION OF PRESSURE ...
Effectiveness Factor, η 2 1 8 6 4 2 0.1 8 6 4 6 8 0.1 Zeroth order First order 2 4 6 8 Asymptotic lines 1 34 2 4 6 8 10 General Thiele Modulus, M T Figure 4.2. Effectiveness factor curves for first order and zeroth order reactions in spherical coordinates. For reactions described by the Langmuir and m-th order rate equations, the curves lie in the narrow band bounded by the first order and zero-th order curves. The dotted line in the band corresponds to m = 0.5 and corresponds approximately to KC s = 1. Correction Function It was shown earlier that in the intermediate range of M T (0.2 < M T < 5), the general asymptotic solution leads to up to -17% error. The error in reaction rate may be amplified to an unacceptably high level when the reaction rate calculation is coupled with the energy equation. Therefore it is desirable to reduce the error in calculating the effectiveness factor by using an empirical correction function with the general asymptotic solution. Two correction functions were constructed to counter the errors associated with the general asymptotic solutions for (a) m-th order rate equations and (b) the Langmuir rate equation, respectively. In order to construct these correction functions, the patterns of error were studied for both the m-th order and the Langmuir rate equations. 2
The resulting observations regarding the patterns of error are shown in Table 4.3. Two observations were made: 1) At a constant value of m or KC s , the maximum error occurs at about M T = 1/2 . Further, as M T departs from 1/2 in a logarithmic scale, the error decreases at approximately the same rate in both directions. That is, if two values of the general modulus (M T1 and M T2) satisfy the following relation: M T 1 1/2 = 1/2 M T2 , (4.16) the error at M T1 is approximately equal to the error at M T2. 2) As the observed reaction order (m obs) in Zone I increases to unity, the error decreases monotonically to zero. In constructing the correction functions, all of the above observations were taken into account. To counter the errors in the whole ranges of M T (from 0 to ∞) and reaction orders (from 0 to 1), two correction functions were constructed as: ⎛ ⎜ fc (MT ,m) = 1 + ⎜ ⎝ 1/2 2 1 2MT + 2 2MT ⎛ 1 fc MT , ⎝ 1+ KCs ⎜ ⎛ ⎞ ⎜ ⎟ = 1 + ⎠ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ 1/2 1 (1−m )2 2 2 1 2MT + 2 2MT These two correction function can be unified into: ⎛ ⎜ fc (MT ,mobs) = 1+ ⎜ ⎝ 1/2 2 1 2MT + 2 2MT 35 ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ 1 2 (1− mobs )2 1 2 (1− 1 ) 1+ KCs 2 where m obs = m for m-th order rate equations, and m obs = 1/(1+KC s) for Langmuir rate equations. (4.17) (4.18) (4.19)
- Page 3 and 4: BRIGHAM YOUNG UNIVERSITY As chair o
- Page 5 and 6: CBK model uses: 1) an intrinsic Lan
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- 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
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- Page 21 and 22: Background 1. Introduction The rate
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- Page 26 and 27: Zone III rate ∝ C og E obs → 0
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- Page 32 and 33: = q obs q max The factor can be use
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- Page 47 and 48: Introduction 4. Analytical Solution
- Page 49 and 50: Task and Methodology Task One of th
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- Page 53: Table 4.1. The Relative Error * (%)
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- Page 59 and 60: Table 4.6. The Relative Error* (%)
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Effectiveness Factor, η<br />
2<br />
1<br />
8<br />
6<br />
4<br />
2<br />
0.1<br />
8<br />
6<br />
4<br />
6 8<br />
0.1<br />
Zeroth order<br />
First order<br />
2 4 6 8<br />
Asymptotic lines<br />
1<br />
34<br />
2 4 6 8<br />
10<br />
General Thiele Modulus, M T<br />
Figure 4.2. Effectiveness factor curves for first order and zeroth order reactions in<br />
spherical coordinates. For reactions described by the Langmuir and m-th<br />
order rate equations, the curves lie in the narrow band bounded by the first<br />
order and zero-th order curves. The dotted line in the band corresponds to<br />
m = 0.5 and corresponds approximately to KC s = 1.<br />
Correction Function<br />
It was shown earlier that in the intermediate range of M T (0.2 < M T < 5), the<br />
general asymptotic solution leads to up to -17% error. The error in reaction rate may be<br />
amplified to an unacceptably high level when the reaction rate calculation is coupled with<br />
the energy equation. Therefore it is desirable to reduce the error in calculating the<br />
effectiveness factor by using an empirical correction function with the general asymptotic<br />
solution. Two correction functions were constructed to counter the errors associated<br />
with the general asymptotic solutions for (a) m-th order rate equations and (b) the<br />
Langmuir rate equation, respectively. In order to construct these correction functions, the<br />
patterns of error were studied for both the m-th order and the Langmuir rate equations.<br />
2