Experimental Quantification of Gas Dispersion in 3D-Printed Logpile Structures Using a Noninvasive Infrared Transmission Technique

3D-printed catalyst structures have the potential to broaden reactor operating windows. However, the hydrodynamic aspects associated with these novel catalyst structures have not yet been quantified in detail. This work applies a recently introduced noninvasive, instantaneous, whole-field concentration measurement technique based on infrared transmission to quantify the rate of transverse gas dispersion in 3D-printed logpile structures. Twenty-two structural variations have been investigated at various operating conditions, and the measured transverse gas dispersion has been correlated to the Péclet number and the structures’ porosity and feature size. It is shown that staggered configurations of these logpile structures offer significantly more tunability of the dispersion behavior compared to straight structures. The proposed correlations can be used to facilitate considerations of reactor design and operating windows.

Magnetic resonance imaging measurements evidence weak dispersion in homogeneous porous media

We measure the dispersion coefficient through homogeneous bead or sand packings at different flow rates from direct magnetic resonance imaging (MRI) visualizations of the transport characteristics of a pulse of paramagnetic nanoparticles. Through two-dimensional imaging we observe homogeneous dispersion inside the sample, but we show that entrance effects may induce significant radial heterogeneities, which would affect the interpretation of the breakthrough curve. Another MRI approach then provides quantitative measurements of the evolution in time of the longitudinal particle distribution in the sample. These data can be analyzed to deduce the coefficient of dispersion independently of entrance effects. The values obtained for this "effective" dispersion coefficient are almost ten times lower than the commonly accepted values.

11-interval PFG pulse sequence for improved measurement of fast velocities of fluids with high diffusivity in systems with short T2(∗)

Magnetic resonance (MR) was used to measure SF6 gas velocities in beds filled with particles of 1.1 mm and 0.5 mm in diameter. Four pulse sequences were tested: a traditional spin echo pulse sequence, the 9-interval and 13-interval pulse sequence of Cotts et al. (1989) and a newly developed 11-interval pulse sequence. All pulse sequences measured gas velocity accurately in the region above the particles at the highest velocities that could be achieved (up to 0.1 ms(-1)). The spin echo pulse sequence was unable to measure gas velocity accurately in the bed of particles, due to effects of background gradients, diffusivity and acceleration in flow around particles. The 9- and 13-interval pulse sequence measured gas velocity accurately at low flow rates through the particles (expected velocity <0.06 ms(-1)), but could not measure velocity accurately at higher flow rates. The newly developed 11-interval pulse sequence was more accurate than the 9- and 13-interval pulse sequences at higher flow rates, but for velocities in excess of 0.1 ms(-1) the measured velocity was lower than the expected velocity. The increased accuracy arose from the smaller echo time that the new pulse sequence enabled, reducing selective attenuation of signal from faster moving nuclei.

Predictions of non-Fickian solute transport in different classes of porous media using direct simulation on pore-scale images

We present predictions of transport through micro-CT images of porous media that include the analysis of correlation structure, velocity, and the dynamics of the evolving plume. We simulate solute transport through millimeter-sized three-dimensional images of a beadpack, a sandstone, and a carbonate, representing porous media with an increasing degree of pore-scale complexity. The Navier-Stokes equations are solved to compute the flow field and a streamline simulation approach is used to move particles by advection, while the random walk method is employed to represent diffusion. We show how the computed propagators (concentration as a function of displacement) for the beadpack, sandstone, and carbonate depend on the width of the velocity distribution. A narrow velocity distribution in the beadpack leads to the least anomalous behavior, where the propagators rapidly become Gaussian in shape; the wider velocity distribution in the sandstone gives rise to a small immobile concentration peak, and a large secondary mobile peak moving at approximately the average flow speed; in the carbonate with the widest velocity distribution, the stagnant concentration peak is persistent, with a slower emergence of a smaller secondary mobile peak, characteristic of highly anomalous behavior. This defines different types of transport in the three media and quantifies the effect of pore structure on transport. The propagators obtained by the model are in excellent agreement with those measured on similar cores in nuclear magnetic resonance experiments by Scheven, Verganelakis, Harris, Johns, and Gladden, Phys. Fluids 17, 117107 (2005).

Robust optimal design of diffusion-weighted magnetic resonance experiments for skin microcirculation

Skin microcirculation plays an important role in several diseases including chronic venous insufficiency and diabetes. Magnetic resonance (MR) has the potential to provide quantitative information and a better penetration depth compared with other non-invasive methods such as laser Doppler flowmetry or optical coherence tomography. The continuous progress in hardware resulting in higher sensitivity must be coupled with advances in data acquisition schemes. In this article, we first introduce a physical model for quantifying skin microcirculation using diffusion-weighted MR (DWMR) based on an effective dispersion model for skin leading to a q-space model of the DWMR complex signal, and then design the corresponding robust optimal experiments. The resulting robust optimal DWMR protocols improve the worst-case quality of parameter estimates using nonlinear least squares optimization by exploiting available a priori knowledge of model parameters. Hence, our approach optimizes the gradient strengths and directions used in DWMR experiments to robustly minimize the size of the parameter estimation error with respect to model parameter uncertainty. Numerical evaluations are presented to demonstrate the effectiveness of our approach as compared to conventional DWMR protocols.

Tracer diffusion in silica inverse opals

We employed fluorescence correlation spectroscopy (FCS) to study the diffusion of small fluorescence tracers in liquid filled silica inverse opals. The inverse opals consisted of a nanoporous silica scaffold spanning a hexagonal crystal of spherical voids of 360 nm diameter connected by circular pores of 70 nm diameter. The diffusion of Alexa Fluor 488 in water and of perylene-3,4,9,10-tetracarboxylic diimide (PDI) in toluene was studied. Three diffusion modes could be distinguished: (1) Free diffusion limited by the geometric constraints given by the inverse opal, where, as compared to the free solution, this diffusion is slowed down by a factor of 3-4, (2) slow diffusion inside the nanoporous matrix of the silica scaffold, and (3) diffusion limited by adsorption. On the length scale of the focus of a confocal microscope of roughly 400 nm diffusion was non-Fickian in all cases.

Flow imaging of fluids in porous media by magnetization prepared centric-scan SPRITE

MRI has considerable potential as a non-destructive probe of porous media, permitting rapid quantification of local fluid content and the possibility of local flow visualization and quantification. In this work we explore a general approach to flow velocity measurement in porous media by combining Cotts pulsed field gradient flow encoding with SPRITE MRI. This technique permits facile and accurate flow and dispersion coefficient mapping of fluids in porous media. This new approach has proven to be robust in characterizing fluid behavior. This method is illustrated through measurements of flow in pipes, flow in sand packs and flow in porous reservoir rocks. Spatially resolved flow maps and local fluid velocity distribution were acquired.

Determining NMR flow propagator moments in porous rocks without the influence of relaxation

Flow propagators, used for the study of advective motion of brine solution in porous carbonate and sandstone rocks, have been obtained without the influence of Nuclear Magnetic Resonance (NMR) relaxation times, T1 and T2. These spin relaxation mechanisms normally result in a loss of signal that varies depending on the displacement zeta of the flowing spins, thereby preventing the acquisition of quantitative propagator data. The full relaxation behaviour of the system under flow needs to be characterised to enable the implementation of a true quantitative measurement. Two-dimensional NMR correlations of zeta-T2 and T1-T2 are used in combination to provide the flow propagators without relaxation weighting. T1-zeta correlations cannot be used due to the loss of T1 information during the displacement observation time Delta. Here the moments of the propagators are extracted by statistical analysis of the full propagator shape. The measured displacements (first moments) are seen to correlate with the expected mean displacements for long observation times Delta. The higher order moments of the propagators determined by this method indicate those obtained previously using a correction were overestimated.

Method for modeling transport of particles in realistic porous networks: application to the computation of NMR flow propagators

We model the transport of particles present in a fluid steadily flowing through a porous medium. The porous medium is described by a representative three-dimensional network. The particles are subjected to advection by the flow and to thermal diffusion. We propose to calculate their trajectories with the continuous time random walk framework. This enables us to efficiently sample disordered networks with realistic topology. The method proposed in this paper is general and can be adapted to model dispersion of tracers. It is applied here to simulate the measurement of the flow propagator [Formula: see text] which is defined as the ensemble density distribution of tracer displacements [Formula: see text], in a given time interval delta t. It can be extracted from pulsed magnetic field gradient spin echo NMR experiments carried out on porous media while fluid is flowing. Preliminary numerical results show good qualitative agreement with experiments.

Auxiliary probe design adaptable to existing probes for remote detection NMR, MRI, and time-of-flight tracing

A versatile, detection-only probe design is presented that can be adapted to any existing NMR or MRI probe with the purpose of making the remote detection concept generally applicable. Remote detection suggests freeing the NMR experiment from the confinement of using the same radio frequency (RF) coil and magnetic field for both information encoding and signal detection. Information is stored during the encoding step onto a fluid sensor medium whose magnetization is later measured in a different location. The choice of an RF probe and magnetic field for encoding can be made based solely on the size and characteristics of the sample and the desired information quality without considering detection sensitivity, as this aspect is dealt with by a separate detector. While early experiments required building probes that included two resonant circuits, one for encoding and one for detection, a modular approach with a detection-only probe as presented here can be used along with any existing NMR probe of choice for encoding. The design of two different detection-only probes is presented, one with a saddle coil for milliliter-sized detection volumes, and the other one with a microsolenoid coil for sub-microliter fluid quantities. As example applications, we present time-of-flight (TOF) tracing of hyperpolarized (129)Xe spins in a gas mixture through coiled tubing using the microsolenoid coil detector and TOF flow imaging through a nested glass container where the gas flow changes its direction twice between inlet and outlet using the saddle coil detector.

Extraction and validation of correlation lengths from interstitial velocity fields using diffusion-weighted MRI

Magnetic Resonance Imaging methods sensitive to individual molecular displacements (q-space MRI) provide a convenient means of measuring dispersion in complex interstitial spaces. Pressure-driven flow experiments through a water-saturated packed bed phantom have been conducted to prove the feasibility of using q-space MRI to measure the coherence length associated with the interstitial velocity field. The method involves measuring the dependence of the apparent dispersion coefficient on the distance along the mean flow by repeating a small number of pulsed-gradient stimulated-echo experiments with increasing gradient pulse separation times. Assuming homogeneous interstitial flow statistics inside the averaging volume, an integral spatial scale characterizing the Eulerian velocity auto-correlation coefficient is extracted via a stochastic convective model. The validity of the a priori statistical description of interstitial flow is verified by comparing with an independent MRI measurement of the Eulerian velocity field using phase contrast methods in the same phantom with pore-level resolution. The integral length scale obtained via q-space MRI agrees with the mean pore size in the present as well as in similar phantoms found in the literature. This method has direct applicability in the quantification of the interstitial morphology of fluid-saturated porous media with resolution independent of voxel size, assuming "perfectly reflecting pore walls" (no surface relaxation) and no contribution to the MR signal from outside the pore space.

A PGSE study of propane gas flow through model porous bead packs

We present a study of the probability density for molecular displacements of gas flowing through bead packs. The three bead packs to be described are composed of polydispersed porous PVC particles, 500 microm glass spheres, and 300 microm polystyrene spheres. A range of velocities (1 cm s(-1) to 1 m s(-1)) and observation times (3-500 ms), hence transport distances, are presented. For comparison we also measure the propagators for water flow in the polystyrene sphere pack. The exchange time between the moving and the stagnant portions of the flow is a strong function of the diffusion coefficient of the fluid. Comparing the propagators between water and propane flowing in similar porous media makes this clear. The gas propagators, for flowing and diffusing molecules, consistently show a feature at the average pore diameter. This feature has previously been observed for similar Peclet number studies in smaller monodispersed bead packs using liquids, but is now demonstrated for larger beads with gas. We analyze and discuss these propagators in the physically intuitive propagator space and also in the well-understood Fourier q space. The extension of NMR PGSE experiments to gas systems allows flow and diffusion information to be obtained over a wider range of length and time scales than with liquids, and also for a new range of physical environments and systems. Interactions between stochastic and deterministic motion are fundamental to the theoretical description of transport in porous media, and the time and length scale dependences are central to an understanding of the resultant dispersive motion.

The NMR microimaging studies of the interplay of mass transport and chemical reaction in porous media

PFG NMR is employed to perform a comparative study of the filtration of water and propane through model porous media. It is shown that the dispersion coefficients for water are dominated by the holdup effects even in a bed of nonporous glass beads. It is demonstrated that correlation experiments such as VEXSY are applicable to gas flow despite the large diffusivity values of gases. The PFG NMR technique is applied to study the gravity driven flow of liquid-containing fine solid particles through a porous bed. The NMR imaging technique is employed to visualize the propagation of autocatalytic waves for the Belousov-Zhabotinsky reaction which is carried out in a model porous medium. It is demonstrated that the wave propagation velocity decreases as the wave crosses the boundary between the bulk liquid and the flooded bead pack. The images detected during the catalytic hydrogenation of alpha-methylstyrene on a single catalyst pellet at elevated temperatures have revealed that the reaction and the accompanying phase transition alter the distribution of the liquid phase within the pellet.

NMR visualization of displacement correlations for flow in porous media

The temporal correlations of velocities for both water and a water-glycerol mixture flowing through a random packings of monodisperse spherical particles have been investigated using two-dimensional nuclear magnetic resonance methods. By combining various flow rates, fluid viscosities, and bead sizes, a wide range of flow parameters has been covered, the dimensionless Peclet number ranging from 100 to 100 000. The velocity exchange spectroscopy (VEXSY) technique has been employed to measure the correlation between velocities during two intervals separated from each other by a mixing time tau(m). This time is made both large and small compared with the time constant tau(c), required for a fluid element possessing the average flow velocity to cover a distance equal to the characteristic size in the system, the bead diameter. The two-dimensional conditional probability of displacement resulting from the VEXSY method reveals the existence of different "subensembles" of molecules, including a slow moving pool whose displacement is dominated by Brownian motion, an intermediate ensemble whose velocities change little over the mixing time, and a fast flowing ensemble which loses correlation due to mechanical dispersion. We find that that the approach to asymptotic dispersion, as tau(c)/tau(m) increases, depends strongly on the Peclet number, the deviation of the velocity autocorrelation function from a monoexponential Ornstein-Uhlenbeck process becoming more pronounced with increasing Peclet number.

Magnetic resonance visualisation of single- and two-phase flow in porous media

Three-dimensional MRI and flow visualisation data are presented for single and two-phase flow occurring within packed beds of glass spheres. The initial motivation for this work has been to understand the operation of fixed-bed reactors used in many chemical processing operations; these systems also serve as model porous media in which to investigate the effect of the structure of a pore space on the flow phenomena occurring within it. For the case of single-phase flow, maps of the liquid shear rate components are calculated from which forces on individual spheres within the bed are obtained. The velocity histogram for flow transverse to the direction of superficial flow is exponential in both negative and positive directions. This form of the velocity histogram implies an exponential form for the displacement propagator, in contrast to the Gaussian distribution obtained by pulsed gradient spin echo measurements. This difference arises because the spatially resolved velocity imaging sequence measures only the average velocity within each voxel and is insensitive to the effects of incoherent (diffusive) motion. Visualisations of air-water flow through a sphere pack are also reported and the capability of MRI to yield information on rivulet formation and surface wetting characteristics is illustrated.

Relationships between flow and NMR relaxation of fluids in porous solids

Measurements are presented which correlate the displacements, X(Delta), determined by PGSE NMR, with the multi-mode transverse 1H NMR relaxation of water flowing through a glass bead pack, for which the dominant relaxation mechanism is diffusion through inhomogeneous internal magnetic fields. Analytical solution for the joint amplitude A[X(Delta), T(2k)] for the case of laminar flow in a circular pipe, with a diffusion-to-surface mechanism, shows that, for other than the lowest mode (k = 0), the contributions to the observed relaxation for a given X(Delta) may involve negative as well as positive amplitudes. The experimental measurements are shown to agree with this general conclusion, showing clear evidence of the presence of relaxation modes with negative amplitude at larger values of X(Delta). It is shown that in these, or similar measurements, which provide a spatially-resolved view of surface-mediated relaxation, allowance must be made for fitting with both positive and negative amplitudes.

Two-dimensional PFG NMR for encoding correlations of position, velocity, and acceleration in fluid transport

A generalized approach to obtain two-dimensional maps of spatial particle coordinates and their derivatives with respect to time by PFG-NMR employing multiple gradient pulses is presented. A sequence of n magnetic field gradient pulses makes it possible, after independent stepping of each pulse and subsequent Fourier transformation, to plot the spin density distribution in coordinate space at n times and along the respective directions of the gradient pulses. In particular, two gradient pulses of effective area k(1) and k(2) separated by a time interval Delta lead to a plot of the combined two-time probability density, W(2)(r(1), 0; r(2), Delta), to find a particle at a coordinate r(1) at t = 0 and at r(2) at t = Delta. A conventional experiment for measuring transport properties by simultaneous stepping of the gradients under the condition k(1) = -k(2) is equivalent to a projection onto the secondary diagonal in the [r(1), r(2)] plot. The main diagonal represents an average position between the two timepoints t = 0 and t = Delta, so that a rotation of the coordinate plot by an angle of 45 degrees allows one to correlate the displacement R = r(2) - r(1) with the averaged position r parallel to the gradient direction. While an average velocity during the time interval Delta can be defined as &vmacr; = R/Delta, an extension toward acceleration and higher order derivatives is straightforward by modification of the pulse sequence. We discuss this concept by application to flow through a circular and a narrowing pipe (confusor), respectively, the experimental results of which are compared to numerical simulations. Copyright 2000 Academic Press.

Correlation time and diffusion coefficient imaging: application to a granular flow system

A parametric method for spatially resolved measurements for velocity autocorrelation functions, R(u)(tau) = <u(t)u(t + tau)>, expressed as a sum of exponentials, is presented. The method is applied to a granular flow system of 2-mm oil-filled spheres rotated in a half-filled horizontal cylinder, which is an Ornstein-Uhlenbeck process with velocity autocorrelation function R(u)(tau) = <u(2)>e(- ||tau ||/tau(c)), where tau(c) is the correlation time and D = <u(2)>tau(c) is the diffusion coefficient. The pulsed-field-gradient NMR method consists of applying three different gradient pulse sequences of varying motion sensitivity to distinguish the range of correlation times present for particle motion. Time-dependent apparent diffusion coefficients are measured for these three sequences and tau(c) and D are then calculated from the apparent diffusion coefficient images. For the cylinder rotation rate of 2.3 rad/s, the axial diffusion coefficient at the top center of the free surface was 5.5 x 10(-6) m(2)/s, the correlation time was 3 ms, and the velocity fluctuation or granular temperature <u(2)> was 1.8 x 10(-3) m(2)/s(2). This method is also applicable to study transport in systems involving turbulence and porous media flows.

NMR characterization of the pore structure and anisotropic self-diffusion in salt water Ice

NMR imaging and one- and two-dimensional self-diffusion propagator measurements of the liquid phase in salt water ice are presented. The properties of the network of brine-filled pores are found to depend on the growth conditions of the ice. Two types of samples are compared: (a) shock-frozen ice produced in the probe in situ and (b) ice grown over several hours under controlled conditions. By shock-freezing, an ice structure could be produced which featured streak-like porous channels of diameters of up to 300 &mgr;m allowing almost unrestricted self-diffusion along one preferential axis but reduced diffusivities in the remaining directions. In ice grown under controlled conditions, the pore sizes are near the resolution limit of the imaging experiment of typically 50 &mgr;m. For this type of samples, strongly non-Gaussian self-diffusion propagators are obtained, indicating restricted self-diffusion on rms scales of 30 &mgr;m. Common to all samples was the observation of highly anisotropic self-diffusion. One- and two-dimensional propagators are compared in order to estimate the degree of anisotropy and the size of the restrictions. Copyright 2000 Academic Press.

Time Correlations in Fluid Transport Obtained by Sequential Rephasing Gradient Pulses

We present a basic experiment by which the evolution of the displacement probability density (propagator) of static or flowing fluid in N successive time intervals is obtained by single labeling, coupled with multiple rephasing events during the course of a pulsed field-gradient sequence. We term this type of sequence SERPENT: SEquential Rephasing by Pulsed field-gradients Encoding N Time-intervals. Realizations of the SERPENT experiment for the case N = 2 which include spin echo, stimulated echo, and Carr-Purcell pulse sequences are suggested. They have in common a spatial spin-labeling of the initial magnetization by a gradient of area q0, followed by successive rephasing via gradients q1 and q2 at times t = Delta1 and t = Delta2, respectively, where q0 + q1 + q2 = 0. A two-dimensional Fourier transform with respect to q1 and q2 gives directly the joint probability density W2(R1, Delta1; R2, Delta2) for displacements R1 and R2 in times Delta1 and Delta2, respectively. q1 and q2 may be in arbitrary directions. Assuming R1 ||R2, the correlation coefficient rhoR1,R2 then reflects the time-history of the fluctuating velocities. The behavior of the cross moment <R1(Delta1). R2(Delta2)> can be obtained from either a full two-dimensional or a set of one-dimensional SERPENT measurements. Experimental results are presented for water flowing through a bed of packed glass beads. While Delta1 is appropriately chosen to sample the short-time velocity field within the system, increasing Delta2 clearly shows the loss of correlation when the average fluid element displacement exceeds the bead diameter. Copyright 1999 Academic Press.

The characterisation of fluid transport in porous solids by means of pulsed magnetic field gradient NMR

The determination, by pulsed field gradient (PFG) nuclear magnetic resonance (NMR), of the probability distributions (propagators) of displacements for fluids undergoing transport by both flow and self-diffusion within porous solids is outlined. The nature of the observed propagator, P delta (Z), for the transport of a single aqueous phase through an outcrop sandstone (Fontainebleau) is described. Recent measurements of the propagators for both aqueous and oil phase flow in the limiting saturation states of irreducible water (Swi) and residual oil (Sor) in the same sample are illustrated through the use of difference propagators. These are shown to emphasise the regions of the propagators most affected by the presence of the second, stationary, phases in these limiting saturation conditions. Measurement of the propagators for both oil and aqueous phases undergoing simultaneous flow are also described for the same sandstone sample and the effect of increasing Swi on the nature of the oil flow is briefly discussed. Finally, a new two-dimensional (2-D) experiment is introduced which measures the propagator P delta (X, Z). This is the joint probability for displacements X and Z in time delta. Some preliminary observations of these two-dimensional propagators are shown for single-phase flow in the Fontainebleau sandstone sample, where Z and X are, respectively, displacements in the axial and radial direction for the cylindrical sample for which the pressure gradient is along Z and where bulk radial flow is constrained to be zero.