WCIPT8 - 8th WORLD CONGRESS ON INDUSTRIAL PROCESS TOMOGRAPHY Iguassu Falls, PR, Brazil, September 26 to 29, 2016 ISIPT - The International Society for Industrial Process Tomography
Design of a sample-based Prior using a phenomenological model for Annular Flow S. P. Pellegrini, J. L. Baliño, F. C. Trigo Departamento de Engenharia Mecânica, Escola Politécnica da Universidade de São Paulo, (Av. Prof. Mello Moraes 2231, 05508-030 São Paulo-SP, Brasil)
[email protected],
[email protected].
ABSTRACT: The characterization of multi-phase flows through electrical impedance tomography (EIT) requires addressing the inherent ill-posed nature of this inverse estimation problem, in order to regularize it. One of the manners to achieve this goal is by using a priori information that suits the system under analysis. For such a purpose, a phenomenological model is created by combining five different correlations from the literature in order to provide a characterization of the three-dimensional phase distribution in horizontal two-phase annular and wet gas flow regimes. First, the model is presented. Then, careful sampling is performed for the input variables of the model and the model is used, along with its expected deviations, to determine a sample of flows, described by their phase distribution in the continuum. Afterwards, the phase distribution is converted to representative values of electrical properties and mapped to a discrete mesh. Finally, this sample of discrete distributions of electrical properties is summarized as a Gaussian density function. This density function shall be used, in a future work, as a prior to solve the inverse EIT problem. The model-based sampling approach for the design of priors presented in this study is especially useful for multiphase flow, given the extensive literature of semi-empirical correlations for different flow patterns. Keywords annular flow, electrical impedance tomography, sample-based prior, wet gas
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INTRODUCTION
Electrical Impedance Tomography (EIT) is a technique which can be used for estimating void fraction in multiphase flows. For that purpose, the electrical properties inside a domain are estimated from measures taken at parts of the domain's boundary and the overall EIT problem results to be inverse, nonlinear and ill-posed (Murai 1985; Holder 2005). One important form of coping with the ill-posed nature of the EIT problem is to introduce known a priori information, thus regularizing the estimation problem. Kaipio and Somersalo (2004, p.70) discuss the generation of priors, in the Bayesian sense, based on a large dataset of values for the distribution of electrical properties. Sample-based priors have been generated from numerical (Santos 2015) and experimental data (Camargo 2011), showing that the use of such priors can provide a less blurred image, more faithful to the actual inhomogeneities of the numerical system analyzed (Camargo 2012). Semi-analytical models are specially relevant in multiphase flows, as many correlations have been computed to describe specific aspects of the flows. These correlations can be combined in phenomenological models, for specific flow patterns. Among the two-phase flow patterns, annular flow is specially relevant for being predominant in industrial applications, as it occurs for a wide range of flow conditions (Collier 1994, p.92). The objective of the present study is to discuss the generation of a sample-based prior using a phenomenological model for two-phase horizontal annular flow.
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WCIPT8 - 8th WORLD CONGRESS ON INDUSTRIAL PROCESS TOMOGRAPHY Iguassu Falls, PR, Brazil, September 26 to 29, 2016 ISPT - The International Society for Industrial Process Tomography
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METHOD
Annular flow is characterized by the presence of a liquid film around the wall of the duct and liquid droplets carried within the core, entrained in the predominant gas phase. If the pipe is not in vertical position (and depending on the relative importance of the gravitational force), the film might be thicker at its lower positions. Additionally, there might be an axial variation in the film thickness, with a traveling wave. These attributes are described by seven parameters: the void fraction, α; the mass and area entrainment factors, e and γ, respectively; the average and bottommost film thickness, ta and tb, respectively; and the frequency and propagation velocity of film waves, fwave and vwave. The mass en-
trainment factor, e, is defined according to (1), as a ratio of the liquid mass flow rate in the core, m
fc
,
f . The entrainment factor, γ, is defined similarly, in terms of cross-sectional area. and the total one, m
e=
m m˙ fc ˙ fc = m˙ f m ˙ fc + m˙ ff
(1)
A phenomenological model is presented in Table 1, with the determination of the output variables listed above in terms of the input variables, being them four fluid properties – liquid viscosity, μf, liquid specific mass, ρf, gas specific mass, ρg, and surface tension, σ, – and three flow conditions – pipe diameter, D, mass flow rate, m , and vapor quality, x. Table 1. Phenomenological model for annular flow Output parameter
Correlation
α
h=− 2 .129 +3. 129
−
() ρ
g f
(
γ
±34.1% (absolute deviation)
(Cioncolini 2012a)
ρ α 1− x g 1− α x ρ f
0
Hypothesis: same velocity for droplets and gas
4t
Geometrical relations
±42.7% (standard deviation)
(Cioncolini 2013)
±25% (absolute deviation)
(Schubring 2008)
±8% (absolute deviation)
(Schubring 2008)
n
()
0 . 8395 cs
−
t t
a b
) 2. 209 −
ρ j2 D c g x+e ( 1− x ) ;ρ = c x σ 1− x +e ρ ρ g f
(1 − α )( 1− γ )=
tb
=
0. 366 Fr
a
√(
(D − t a )
D2 1 . 45
1+0 .366 Fr
1. 45
ρ v2 c c ρ − ρ gD f c Gx √Fr mod f =0. 035 wave ρ D g ρ ( Gx ) / ρ g g Gx Fr = = mod ρ ρ √gD l √gD l Fr=
2
(Cioncolini 2012b)
hx n
1+( h − 1 ) x 0 .2186 ρ 0 . 5150 g ; n=0 . 3487+0 . 6513 ρ f
γ=e
ta
vwave
±2.9% (standard deviation)
e= 1+279 . 6 We We = cs
fwave
Reference
α= ρ
e
Deviation
v
wave
)
G √x − 0 .25 Re G ρ g GD Re = G μ l
=0. 42
WCIPT8 - 8th WORLD CONGRESS ON INDUSTRIAL PROCESS TOMOGRAPHY Iguassu Falls, PR, Brazil, September 26 to 29, 2016 ISIPT - The International Society for Industrial Process Tomography
Additionally, a compilation of experimental data (Westende 2008) is used to determine the diameter of the liquid droplets entrained in the gas core, as shown by the probability density function shown in (2), which describes the statistics of the droplets as a function of the diameter dd.
PDF d =
{
0 . 02443 exp (− 0 . 02 d d ) if 10 μm
