Abstract
The conjugate gradient method, or iterative regularization method, based inverse algorithm is utilized in this work to predict the unknown concentration-dependent reaction rate function for an annular-bed reactor (ABR) using interior measurements of concentration distributions. Since no prior information on the functional form of unknown reaction rate is available, it can be classified as function estimation for the inverse calculation. The validity and accuracy of this inverse ABR problem are examined using the simulated exact and inexact concentration measurements in the numerical experiments. Results show that the estimation of the concentration-dependent reaction rate function can be obtained within a very short CPU time on an Intel Xeon Core 2 2.00 GHz personal computer, and reliable estimations can still be obtained when measurement errors are considered.
Acknowledgment
This work was supported in part through the National Science Council, ROC, Grant number, NSC-100-2221-E-006-011-MY3.
Nomenclature
- Am, An
aspect ratios
- D
dimensionless radial dispersion coefficient
- dp
diameter of the inner core packing
- ERR
relative errors
- f
estimated dimensionless concentration
- J
functional defined by eq. (3)
- J′
gradient of functional defined by eq. (14)
- L
reactor length (cm)
- P
direction of descent defined by eqs (5a) and (5b)
- Re
Reynolds number
- R(f)
dimensionless reaction rate
- r
dimensionless coordinate
- rC–SC
dimensionless radius of the core–screen interface
- rSC–B
dimensionless radius of the screen–bed interface
radius of the inside of the tube wall (cm)
- Sc
Schmidt number
- Sr
Ratio of local velocity to cross-sectional average velocity
- Y
measured dimensionless concentration
- z
dimensionless coordinate
- Greek letters
- β
search step sizes
- γ
conjugate coefficients
- λ
Lagrange multipliers defined by eq. (12)
sensitivity functions defined by eq. (7)
- η
convergence criteria
- σ
standard deviation for measurement
- φ
Thiele modulus
- ω
random number
- Superscript
- B
for the bed region
- C
for the core region
- n
iteration index
- SC
for the screen region
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