Abstract
A conjugate gradient method (CGM) based on the inverse algorithm is used to estimate the unknown fouling-layer profile on the inner wall of a pipe system using simulated temperature measurements taken within the pipe wall. It is assumed that no prior information is available about the functional form of the unknown profile. Therefore, the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The accuracy of the inverse analysis is examined using the simulated exact and inexact temperature measurements. The results show that the excellent estimation of the fouling-layer profile can be obtained for the test case considered in this study. The technique presented in this study can be used in a warning system to call for pipe maintenance when the thickness of fouling exceeds a predefined criterion.
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Abbreviations
- h :
-
convection heat transfer coefficient W·m−2·K−1
- J :
-
functional
- J′:
-
gradient of functional
- k :
-
thermal conductivity, W·m−1·K−1
- L :
-
length of the pipe, m
- M :
-
total number of measuring positions
- p :
-
direction of descent
- R 1 :
-
inner radius of the pipe, m
- R 2 :
-
outer radius of the pipe, m
- r :
-
spatial radial coordinate, m
- T :
-
temperature, K
- T in :
-
inlet temperature, K
- T ∞ :
-
ambient temperature, K
- u x :
-
axial fluid velocity, m·s−1
- x :
-
spatial axial coordinate, m
- Y :
-
measurement temperature, K
- Δ:
-
small variation quality
- α :
-
thermal diffusivity, m2·s−1
- β :
-
step size
- γ :
-
conjugate coefficient
- η :
-
very small value
- λ :
-
variable used in the adjoint problem
- σ :
-
standard deviation
- \( \varpi \) :
-
random variable
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Project supported by the National Science Council of Taiwan of China (No. NSC 97-2221-E-168-039)
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Chen, Wl., Yang, Yc., Lee, Hl. et al. Inverse estimation for unknown fouling geometry on inner wall of forced-convection pipe. Appl. Math. Mech.-Engl. Ed. 32, 55–68 (2011). https://doi.org/10.1007/s10483-011-1393-9
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DOI: https://doi.org/10.1007/s10483-011-1393-9