Optimising magnetic resonance sampling patterns for parametric characterisation.

Sampling strategies are often central to experimental design. Choosing efficiently which data to acquire can improve the estimation of parameters and reduce the acquisition time. This work is focused on designing optimal sampling patterns for Nuclear Magnetic Resonance (NMR) applications, illustrated with respect to the best estimate of the parameters characterising a lognormal distribution. Lognormal distributions are commonly used as fitting models for distributions of spin-lattice relaxation time constants, spin-spin relaxation time constants and diffusion coefficients. A method for optimising the choice of points to be sampled is presented which is based on the Cramér-Rao Lower Bound (CRLB) theory. The method's capabilities are demonstrated experimentally by applying it to the problem of estimating the emulsion droplet size distribution from a pulsed field gradient (PFG) NMR diffusion experiment. A difference of <5% is observed between the predictions of CRLB theory and the PFG NMR experimental results. It is shown that CLRB theory is stable down to signal-to-noise ratios of ∼10. A sensitivity analysis for the CRLB theory is also performed. The method of optimizing sampling patterns is easily adapted to distributions other than lognormal and to other aspects of experimental design; case studies of optimising the sampling scheme for a fixed acquisition time and determining the potential for reduction in acquisition time for a fixed parameter estimation accuracy are presented. The experimental acquisition time is typically reduced by a factor of 3 using the proposed method compared to a constant gradient increment approach that would usually be used

Development of ultrafast UTE imaging for granular systems

Ultrashort echo time (UTE) imaging is commonly used in medical MRI to image 'solid' types of tissue; to date it has not been widely used in engineering or materials science, in part due to the relatively long imaging times required. Here we show how the acquisition time for UTE can be reduced to enable a preliminary study of a fluidized bed, a type of reactor commonly used throughout industry containing short T$_{2}$$^{∗}$ material and requiring fast imaging. We demonstrate UTE imaging of particles with a T$_{2}$$^{∗}$ of only 185μs, and an image acquisition time of only 25ms. The images are obtained using compressed sensing (CS) and by exploiting the Hermitian symmetry of k-space, to increase the resolution beyond that predicted by the Nyquist theorem. The technique is demonstrated by obtaining one- and two-dimensional images of bubbles rising in a model fluidized bed reactor.HTF would like to acknowledge the financial support of the Gates Cambridge Trust. All authors would like to acknowledge the financial support of the EPSRC (EP/K008218/1, EP/F047991/1 and EP/K039318/1)

Obtaining sparse distributions in 2D inverse problems

The mathematics of inverse problems has relevance across numerous estimation problems in science and engineering. L1 regularization has attracted recent attention in reconstructing the system properties in the case of sparse inverse problems; i.e., when the true property sought is not adequately described by a continuous distribution, in particular in Compressed Sensing image reconstruction. In this work, we focus on the application of L1 regularization to a class of inverse problems; relaxation-relaxation, T1–T2, and diffusion-relaxation, D–T2, correlation experiments in NMR, which have found widespread applications in a number of areas including probing surface interactions in catalysis and characterizing fluid composition and pore structures in rocks. We introduce a robust algorithm for solving the L1 regularization problem and provide a guide to implementing it, including the choice of the amount of regularization used and the assignment of error estimates. We then show experimentally that L1 regularization has significant advantages over both the Non-Negative Least Squares (NNLS) algorithm and Tikhonov regularization. It is shown that the L1 regularization algorithm stably recovers a distribution at a signal to noise ratio < 20 and that it resolves relaxation time constants and diffusion coefficients differing by as little as 10%. The enhanced resolving capability is used to measure the inter and intra particle concentrations of a mixture of hexane and dodecane present within porous silica beads immersed within a bulk liquid phase; neither NNLS nor Tikhonov regularization are able to provide this resolution. This experimental study shows that the approach enables discrimination between different chemical species when direct spectroscopic discrimination is impossible, and hence measurement of chemical composition within porous media, such as catalysts or rocks, is possible while still being stable to high levels of noise.A.R. acknowledges Gates Trust Cambridge for financial support. A.J.S. and L.F.G. would like to acknowledge support from EPSRC (EP/N009304/1)

Magnetic Resonance Imaging and Velocity Mapping in Chemical Engineering Applications.

This review aims to illustrate the diversity of measurements that can be made using magnetic resonance techniques, which have the potential to provide insights into chemical engineering systems that cannot readily be achieved using any other method. Perhaps the most notable advantage in using magnetic resonance methods is that both chemistry and transport can be followed in three dimensions, in optically opaque systems, and without the need for tracers to be introduced into the system. Here we focus on hydrodynamics and, in particular, applications to rheology, pipe flow, and fixed-bed and gas-solid fluidized bed reactors. With increasing development of industrially relevant sample environments and undersampling data acquisition strategies that can reduce acquisition times to <1 s, magnetic resonance is finding increasing application in chemical engineering research

Retaining both discrete and smooth features in 1D and 2D NMR relaxation and diffusion experiments.

A new method of regularization of 1D and 2D NMR relaxation and diffusion experiments is proposed and a robust algorithm for its implementation is introduced. The new form of regularization, termed the Modified Total Generalized Variation (MTGV) regularization, offers a compromise between distinguishing discrete and smooth features in the reconstructed distributions. The method is compared to the conventional method of Tikhonov regularization and the recently proposed method of L1 regularization, when applied to simulated data of 1D spin-lattice relaxation, T1, 1D spin-spin relaxation, T2, and 2D T1-T2 NMR experiments. A range of simulated distributions composed of two lognormally distributed peaks were studied. The distributions differed with regard to the variance of the peaks, which were designed to investigate a range of distributions containing only discrete, only smooth or both features in the same distribution. Three different signal-to-noise ratios were studied: 2000, 200 and 20. A new metric is proposed to compare the distributions reconstructed from the different regularization methods with the true distributions. The metric is designed to penalise reconstructed distributions which show artefact peaks. Based on this metric, MTGV regularization performs better than Tikhonov and L1 regularization in all cases except when the distribution is known to only comprise of discrete peaks, in which case L1 regularization is slightly more accurate than MTGV regularization

PFG NMR and Bayesian analysis to characterise non-Newtonian fluids

Many industrial flow processes are sensitive to changes in the rheological behaviour of process fluids, and there therefore exists a need for methods that provide online, or inline, rheological characterisation necessary for process control and optimisation over timescales of minutes or less. Nuclear magnetic resonance (NMR) offers a non-invasive technique for this application, without limitation on optical opacity. We present a Bayesian analysis approach using pulsed field gradient (PFG) NMR to enable estimation of the rheological parameters of Herschel-Bulkley fluids in a pipe flow geometry, characterised by a flow behaviour index n, yield stress τ$_{0}$, and consistency factor k, by analysis of the signal in q-space. This approach eliminates the need for velocity image acquisition and expensive gradient hardware. We investigate the robustness of the proposed Bayesian NMR approach to noisy data and reduced sampling using simulated NMR data and show that even with a signal-to-noise ratio (SNR) of 100, only 16 points are required to be sampled to provide rheological parameters accurate to within 2% of the ground truth. Experimental validation is provided through an experimental case study on Carbopol 940 solutions (model Herschel-Bulkley fluids) using PFG NMR at a $^{1}$H resonance frequency of 85.2MHz; for SNR>1000, only 8 points are required to be sampled. This corresponds to a total acquisition time of <60s and represents an 88% reduction in acquisition time when compared to MR flow imaging. Comparison of the shear stress-shear rate relationship, quantified using Bayesian NMR, with non-Bayesian NMR methods demonstrates that the Bayesian NMR approach is in agreement with MR flow imaging to within the accuracy of the measurement. Furthermore, as we increase the concentration of Carbopol 940 we observe a change in rheological characteristics, probably due to shear history-dependent behaviour and the different geometries used. This behaviour highlights the need for online, or inline, rheological characterisation in industrial process applications.AJS and LFG wish to thank the EPSRC (Grant numbers EP/F047991/1 and EP/K039318/1) and TWB wishes to thank the EPSRC and Johnson Matthey plc for financial support

Magnetic resonance velocity imaging of gas flow in a diesel particulate filter

Magnetic resonance (MR) velocity imaging has been used to investigate the gas flow in a diesel particulate filter (DPF), with sulphur hexafluoride (SF$_{6}$) being used as the MR-active gas. Images of the axial velocity were acquired at ten evenly spaced positions along the length of the filter, for three flow conditions corresponding to Reynolds number of Re = 106, 254 and 428 in the filter channels. From the velocity images, averaged axial and through-wall velocity, as a function of position along the length of the filter, have been obtained. These experimentally obtained velocity profiles are analysed and a qualitative comparison with the results of previously reported numerical simulations is made. The MR measurements were used in subsequent analysis to quantify the uniformity of the through-wall velocity profiles. From this it was observed that for higher Re flows, the through-wall velocity profile became less uniform, and the implications that this has on particulate matter deposition are discussed. The MR technique demonstrated herein provides a useful method to advance our understanding of hydrodynamics and mass transfer within DPFs and also for the validation of numerical simulations used in their design and optimization.NPR acknowledges the EPSRC and Johnson Matthey for a CASE award. LFG and AJS also wish to thank EPSRC for financial support (EP/K039318/1)

Investigation of Void Fraction Schemes for Use with CFD-DEM Simulations of Fluidized Beds

© 2018 American Chemical Society. This paper investigates the spatial resolution of computational fluid dynamics-discrete element method (CFD-DEM) simulations of a bubbling fluidized bed for seven different void fraction schemes. Fluid grids with cell sizes of 3.5, 1.6, and 1.3 particle diameters were compared. The particle velocity maps from all of the void fraction schemes were in good qualitative agreement with the experimental data collected using magnetic resonance imaging (MRI). Refining the fluid grid improved the quantitative agreement due to a more accurate representation of flow near the gas distributor. The approach proposed by Khawaja et al. [ J. Comput. Multiphase Flows 2012, 4, 183-192 ] provided the closest match to the exact void fraction though only the particle centered method differed significantly. These results indicate that the fluid grid used for CFD-DEM simulations must be sufficiently fine to represent the inlet flow realistically and that a void fraction scheme such as that proposed by Khawaja be used

Measuring velocity and turbulent diffusivity in wall-flow filters using compressed sensing magnetic resonance

Gas-phase compressed sensing magnetic resonance methods have been used to image gas flow velocity and turbulent diffusivity in wall-flow particulate filters. Two-dimensional magnetic resonance velocity imaging was used to observe the local distribution of gas velocity in the direction of superficial flow (z) in the entrance and exit regions of the filter at an in-plane spatial resolution of 140 µm (x) × 140 µm (y) and 140 µm (x) × 390 µm (z) perpendicular to and parallel with the direction of superficial flow, respectively. Images were acquired in 14 min. Three-dimensional images of the turbulent diffusivity were acquired at a spatial resolution of 280 µm (x) × 280 µm (y) × 1250 µm (z) for channel Reynolds numbers, Rec, of 210, 360, 720 and 1140. These data provide evidence of regions of turbulence inside the filter that has not been predicted by earlier numerical simulations. For Rec = 1140, a three-dimensional velocity image was also obtained at the same spatial resolution as the image of turbulent diffusivity; the data acquisition time was 2 h. Co-registration of these two images enables visualisation of the spatial extent and magnitude of these two characteristics of the flow field.JDC would like to thank Johnson Matthey and the EPSRC for a CASE award (award reference 1628588)

Accelerating the estimation of 3D spatially resolved T2 distributions.

Obtaining quantitative, 3D spatially-resolved T2 distributions (T2 maps) from magnetic resonance data is of importance in both medical and porous media applications. Due to the long acquisition time, there is considerable interest in accelerating the experiments by applying undersampling schemes during the acquisition and developing reconstruction techniques for obtaining the 3D T2 maps from the undersampled data. A multi-echo spin echo pulse sequence is used in this work to acquire the undersampled data according to two different sampling patterns: a conventional coherent sampling pattern where the same set of lines in k-space is sampled for all equally-spaced echoes in the echo train, and a proposed incoherent sampling pattern where an independent set of k-space lines is sampled for each echo. The conventional reconstruction technique of total variation regularization is compared to the more recent techniques of nuclear norm regularization and Nuclear Total Generalized Variation (NTGV) regularization. It is shown that best reconstructions are obtained when the data acquired using an incoherent sampling scheme are processed using NTGV regularization. Using an incoherent sampling pattern and NTGV regularization as the reconstruction technique, quantitative results are obtained at sampling percentages as low as 3.1% of k-space, corresponding to a 32-fold decrease in the acquisition time, compared to a fully sampled dataset