CFD-DEM modelling and simulation of pneumatic conveying: A review

Direct measurements of many properties of real-world systems are not possible. Information on these properties can only be inferred from the result of measurements of other quantities which may be measured directly. The process comprising direct measurements of certain characteristics of the object followed by inference on its sought-for properties from the directly measured characteristics based on a mathematical relation between unknown properties and measured characteristics is called indirect measurement, whereas inference is referred to as an inverse problem in indirect measurement. In general an inverse problem consists either in determining the characteristics of a system under study, driven by controlled or known exciting signals, or in reconstructing exciting signals acting on a system whose internal characteristics are known. In both cases, it is formulated in terms of a mathematical model relating unknown and measured characteristics and signals. One can distinguish continuous and discrete inverse problems, depending on whether the measured and sought-for quantities are represented by functions or by vectors (tuples), respectively. Very many nontrivial inverse problems in indirect measurements are ill-posed which means that they have no solution or the solution exists but is non-unique or unstable, i.e. very small disturbances in the measurement data result in large disturbances in the result of inference. High error amplification is referred to as ill-conditioning. Ill-posedness and ill-conditioning result from the lack of information on sought-for quantities, carried by the measurement data. Therefore, a priori knowledge about the space of admissible solutions has to be employed for solving such inverse problems. The theory of inverse problems and – in particular – effective numerical methods for solving them are of great importance for measurement science and technology; they are crucial for the development of many measurement, imaging and diagnostic techniques. Indirect measurements may be formulated using various mathematical models of the measurement object followed by a measuring system. A broad class of inverse problems, being of importance for indirect measurements, is formulated in terms of Fredholm integral equations of the first kind. These problems are ill-posed and strongly ill-conditioned after discretization. Therefore, sophisticated inverse procedures, utilizing various kinds of a priori knowledge, are applied for solving them. In this paper, theoretical and numerical aspects of inverse problem in indirect measurements are reviewed. In particular the concept of generalized solution (pseudosolution) and the notion of well-posedness is presented and analysed. The review is focused on inverse problems formulated in terms of Fredholm integral equations of the first kind: a general presentation of such problems, at the level of functional analysis, is followed by an overview of numerical aspects of their discretized versions. A concise presentation of selected groups of numerical methods, called inverse methods, for solving inverse problems is also provided.

/pdf/review-article-inverse-problems-formulated-in-terms-of-first-iwz5l3g7mp.pdf

Review article inverse problems formulated in terms of first-kind fredholm integral equations in indirect measurements

Direct measurements of many properties of real-world systems are not possible Information on these properties can only be inferred from the result of measurements of other quantities which may be measured directly The process comprising direct measurements of certain characteristics of the object followed by inference on its sought-for properties from the directly measured characteristics based on a mathematical relation between unknown properties and measured characteristics is called indirect measurement, whereas inference is referred to as an inverse problem in indirect measurement In general an inverse problem consists either in determining the characteristics of a system under study, driven by controlled or known exciting signals, or in reconstructing exciting signals acting on a system whose internal characteristics are known In both cases, it is formulated in terms of a mathematical model relating unknown and measured characteristics and signals One can distinguish continuous and discrete inverse problems, depending on whether the measured and sought-for quantities are represented by functions or by vectors (tuples), respectively Very many nontrivial inverse problems in indirect measurements are ill-posed which means that they have no solution or the solution exists but is non-unique or unstable, ie very small disturbances in the measurement data result in large disturbances in the result of inference High error amplification is referred to as ill-conditioning Ill-posedness and ill-conditioning result from the lack of information on sought-for quantities, carried by the measurement data Therefore, a priori knowledge about the space of admissible solutions has to be employed for solving such inverse problems The theory of inverse problems and – in particular – effective numerical methods for solving them are of great importance for measurement science and technology; they are crucial for the development of many measurement, imaging and diagnostic techniques Indirect measurements may be formulated using various mathematical models of the measurement object followed by a measuring system A broad class of inverse problems, being of importance for indirect measurements, is formulated in terms of Fredholm integral equations of the first kind These problems are ill-posed and strongly ill-conditioned after discretization Therefore, sophisticated inverse procedures, utilizing various kinds of a priori knowledge, are applied for solving them In this paper, theoretical and numerical aspects of inverse problem in indirect measurements are reviewed In particular the concept of generalized solution (pseudosolution) and the notion of well-posedness is presented and analysed The review is focused on inverse problems formulated in terms of Fredholm integral equations of the first kind: a general presentation of such problems, at the level of functional analysis, is followed by an overview of numerical aspects of their discretized versions A concise presentation of selected groups of numerical methods, called inverse methods, for solving inverse problems is also provided

Inverse problems formulated in terms of first-kind Fredholm integral equations in indirect measurements

By means of a numerical study we show particle-size distributions retrieved with the Chin-Shifrin, Phillips-Twomey, and singular value decomposition methods. Synthesized intensity data are generated using Mie theory, corresponding to unimodal normal, gamma, and lognormal distributions of spherical particles, covering the size parameter range from 1 to 250. Our results show the advantages and disadvantages of each method, as well as the range of applicability for the Fraunhofer approximation as compared to rigorous Mie theory.

Reconstruction of particle-size distributions from light-scattering patterns using three inversion methods.

A polar nephelometer that measures the light scattered into a two dimensional plane by a stream of aerosols that intersect a 350 milliwatt diode laser beam with a wavelength of 670 nm is described. A half wave plate is used to orient the laser light both parallel and perpendicular to the measurement plane. This scattering information combined with simultaneously measured size distribution information is used to determine the real (mr) refractive index of the aerosols using a Genetic Algorithm search method. Errors in the retrieved mr based on noise in the scattering measurement as well as uncertainties in the size distribution measurement are examined. The mr of polystyrene latex spheres and ammonium sulfate droplets are determined and match expectations within experimental uncertainties using Mie-Lorenz theory. The angular scattering properties of spherical secondary organic aerosols generated by oxidizing α -pinene do not follow this theory and calls into question the inferred mr.

https://www.tandfonline.com/doi/pdf/10.1080/02786820701432640

Genetic algorithm inversion of dual polarization polar nephelometer data to determine aerosol refractive index

A stochastic inverse technique based on a genetic algorithm (GA) to invert particle-size distribution from angular light-scattering data is developed. This inverse technique is independent of any given a priori information of particle-size distribution. Numerical tests show that this technique can be successfully applied to inverse problems with high stability in the presence of random noise and low susceptibility to the shape of distributions. It has also been shown that the GA-based inverse technique is more efficient in use of computing time than the inverse Monte Carlo method recently developed by Ligon et al. [Appl. Opt.35, 4297 (1996)].

Yong Lu

Papers

Inversion of particle-size distribution from angular light-scattering data with genetic algorithms

Experimental and numerical study of particle velocity distribution in the vertical pipe after a 90° elbow

Investigation of Flow Patterns of Gas-Solid Granular Flow over Horizontal Pipe Cross-sections by FLUENT & EDEM