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At least 1,171 records · Page 65

Effect of Feed Composition on Cold-Cap Formation in Laboratory-Scale Melter - 16336

The development of advanced glass formulations is part of the plan for reducing the cost and time for treatment and vitrification of the 210,000 m{sup 3} of nuclear waste at the Hanford Site in southeastern Washington State. One property of interest in this development is melt viscosity, which has a decisive influence on the rate of glass production. In an electric melter, the conversion process from feed-to-glass above the melt pool occurs in the cold cap. At the final stage of conversion, when the glass-forming melt becomes connected, gas-evolving reactions cause foaming. The transient melt viscosity affects foam stability. Three glasses were formulated with viscosities of 1.5, 3.5, and 9.5 Pa s at 1150 deg. C by varying the SiO{sub 2} content at the expense of B{sub 2}O{sub 3}, Li{sub 2}O, and Na{sub 2}O kept at constant proportions. Cold caps were produced by charging simulated high-alumina, high-level waste feeds in a laboratory-scale melter. The spread of the feed on the cold cap during charging and the cross-sectional structure of the final cold caps were compared. The amount of the foam and the size of the bubbles increased as the viscosity increased. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

Impact of viscous fingering and permeability heterogeneity on fluid mixing in porous media

Fluid mixing plays a fundamental role in many natural and engineered processes, including groundwater flows in porous media, enhanced oil recovery, and microfluidic lab-on-a-chip systems. Recent developments have explored the effect of viscosity contrast on mixing, suggesting that the unstable displacement of fluids with different viscosities, or viscous fingering, provides a powerful mechanism to increase fluid-fluid interfacial area and enhance mixing. However, existing studies have not incorporated the effect of medium heterogeneity on the mixing rate. Here, we characterize the evolution of mixing between two fluids of different viscosity in heterogeneous porous media. We focus on a practical scenario of divergent-convergent flow in a quarter five spot geometry prototypical of well-driven groundwater flows. We study by means of numerical simulations the impact of permeability heterogeneity and viscosity contrast on the breakthrough curves and mixing efficiency, and we rationalize the nontrivial mixing behavior that emerges from the competition between the creation of fluid-fluid interfacial area and channeling.

58 GEOSCIENCES↗

Dominant Controls of Downdip Afterslip and Viscous Relaxation on the Postseismic Displacements Following the M w 7.9 Gorkha, Nepal, Earthquake

Here, we analyze three-dimensional GPS coordinate time series from continuously operating stations in Nepal and South Tibet and calculate the initial 1 year postseismic displacements. We first investigate models of poroelastic rebound, afterslip, and viscoelastic relaxation individually and then attempt to resolve the trade-offs between their contributions by evaluating the misfit between observed and simulated displacements. We compare kinematic inversions for distributed afterslip with stress-driven afterslip models. The modeling results show that no single mechanism satisfactorily explains near- and far-field postseismic deformation following the Gorkha earthquake. When considering contributions from all three mechanisms, we favor a combination of viscoelastic relaxation and afterslip alone, as poroelastic rebound always worsens the misfit. The combined model does not improve the data misfit significantly, but the inverted afterslip distribution is more physically plausible. The inverted afterslip favors slip within the brittle-ductile transition zone downdip of the coseismic rupture and fills the small gap between the mainshock and largest aftershock slip zone, releasing only 7% of the coseismic moment. Our preferred model also illuminates the laterally heterogeneous rheological structure between India and the South Tibet. The transient and steady state viscosities of the upper mantle beneath Tibet are constrained to be greater than 10 18 Pa s and 10 19 Pa s, whereas the Indian upper mantle has a high viscosity ≥10 20 Pa s. The viscosity in the lower crust of southern Tibet shows a clear trade-off with its southward extent and thickness, suggesting an upper bound value of ~8 × 10 19 Pa s for its steady state viscosity.

58 GEOSCIENCES↗

Synthesis and characterization of PANI/P-2M conductive composites: Thermal, rheological, mechanical, and electrical properties

In this article, a new conductive thermosetting resin system was developed using Polyaniline with 2-Methacryloyloxyethyl acid phosphate (P-2 M) as a dopant. The resin has shown better steady state viscosity and enhanced mechanical properties as high as 3±0.8 GPa as compared to our previously developed resin system of PANI-DBSA. A DSC analysis was conducted to estimate the activation energy and the exact amount of energy for doping and curing process of the resin. The resin was synthesized with various wt% of PANI and each combination was characterized in detail. As the amount of PANI was increased, the electrical conductivity significantly enhanced. The maximum electrical conductivity obtained is 0.5 S/m, which is lower than other conductive resins but is many orders higher than epoxy resin. The uncured resin was studied for its viscosity change with shear rate and the constitution of PANI in the resin and we obtained a desirable steady-state viscosity of the resin. Thus, the newly developed resin with efficient steady state viscosity in the favorable range, optimized electrical conductivity, and superior mechanical properties, can be applied into CFRP fabrication.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

In-Process Melt Separation of Depolymerized PET in PET/PE Blends for Upcycling via Twin-Screw Extrusion

Our previous work focused on depolymerizing polyethylene terephthalate (PET) in twin-screw extrusion, as part of a broader project to continuously separate PET from polyolefins in the melt. This study focused on linear low-density polyethylene (LLDPE) and PET films and the use of ethylene glycol (EG), diethylene glycol (DEG), triethylene glycol (TEG), and bis(2-hydroxyethyl) terephthalate (BHET) to depolymerize the PET in the extruder to levels above 90% Mw. In this work, the focus will shift to achieving separation of the two polymers in the twin-screw extruder, which is made possible due to a 90% reduction in the Mw of the PET, which caused a decrease of its viscosity by several orders of magnitude. Owing to the viscosity difference and pressure buildup in the die, the low-viscosity PET preferentially exited a degassing vent instead of going through the die. This is because the flow of the PET would travel through a non-pressure vent rather than through a high-pressure die. However, owing to the higher viscosity of the LLDPE, the pressure was too high to pass through such a small diameter vent hole. Supercritical CO 2 (SCCO 2 ) was used to assist in this extraction, but SCCO 2 negatively impacted the overall degree of separation. Through analysis of the separated materials, it was concluded that a high separation of the two materials was achieved. TGA and FTIR confirmed that the material separated from the vent was 100% PET. The material removed from the die was composed of 95% LLDPE and 5% PET.

36 MATERIALS SCIENCE↗

In‐Process Melt Separation of PE/PET Blends for Upcycling via Twin‐Screw Extrusion. Impact of Catalyst Reagents on PET Depolymerization

Part 1 of this work focused on depolymerizing polyethylene terephthalate (PET) in twin-screw extrusion, as part of a broader project to continuously separate PET from polyolefins in the melt. This study focused on linear low-density polyethylene (LLDPE) and PET films and the use of ethylene glycol (EG), diethylene glycol (DEG), triethylene glycol (TEG), and bis(2-hydroxyethyl) terephthalate (BHET) to depolymerize the PET in the extruder to levels above 90% Mw. In part 2, the focus will shift to achieving separation of the two polymers in the twin-screw extruder, which is made possible due to a 90% reduction in the Mw of the PET, which caused a decrease of its viscosity by several orders of magnitude. Owing to the viscosity difference and pressure buildup in the die, the low-viscosity PET preferentially exited a degassing vent instead of going through the die. This is because the flow of the PET would travel through a non-pressure vent rather than through a high-pressure die. However, owing to the higher viscosity of the LLDPE, the pressure was too high to pass through such a small diameter vent hole. Supercritical CO₂ (SCCO₂) was used to assist in this extraction, but SCCO₂ negatively impacted the overall degree of separation. Through analysis of the separated materials, it was concluded that a very high separation of the two materials was achieved. TGA and FTIR confirmed that the material separated from the vent was 100% PET. The material removed from the die was composed of 90% LLDPE and 10% PET.

36 MATERIALS SCIENCE↗

Dynamic, hollow nanotubular networks with superadjustable pH-responsive and temperature resistant rheological characteristics

Recently, the interest in stimuli-responsive and adaptable materials has continuously grown in various fields and applications. For such responsive systems, different triggers, including pH, light, pressure, temperature, and electric field, have been utilized to control dynamics and assembly. Among these, pH is one of the most convenient, energy-efficient, and economic modalities. Besides, plenty of traditional materials have poor thermal and salt stability, limiting their applications. Herein, we report a new design of a pH-responsive viscoelastic supramolecular complex (VSC) based on commplexation of a new long-chain amino-amide and maleic acid. In this study, the system demonstrated a sol–gel-sol transition from pH 2 to 10, with the largest static viscosity occurring at pH 6 (~1000 Pa·s) and the smallest viscosity at pH 4 (~3.3 Pa·s), indicating ~ 300-fold control over the viscosity. For a given concentration, the static viscosity of VSC was about 15 times larger than that of CTAB/NaSal, a well-established dynamic viscoelastic system, and no pH-responsiveness was observed for the traditional system. In addition, the VSC demonstrated a superior temperature tolerance and lower temperature dependence. The potential of these intriguing dynamics viscoelastic systems was evaluated for hydraulic fracturing and enhanced oil recovery applications. Proppant settling velocity of DMAA/MA was 500 ~ 1000 times lower than that of CTAB/NaSal and common traditional polymers. Likewise, the oil recovery percentage could be significantly improved with the utilization of DMAA/MA compared to the CTAB/NaSal (86 % vs 52 %). Aside from applications in hydraulic fracturing and enhanced oil recovery, we anticipate that the intriguing rheological properties of this viscoelastic system can be beneficial for other chemical engineering applications including personal care products, cosmetics, lubricants, and biomedical gels.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Migration and accumulation of hydrous mantle incipient melt in the Earth's asthenosphere: Constraints from in-situ falling sphere viscometry measurements

The Earth's oceanic lithosphere-asthenosphere boundary (LAB) is marked by a notable decrease in seismic wave velocities and an increase in attenuation. This phenomenon is likely attributed to the accumulation of partial melt at the top of the asthenosphere. Nevertheless, the process involving the upward migration and aggregation of low-degree partial melts, highly likely to be mantle incipient melts in the asthenosphere, remains underexplored. Viscosity is a key factor controlling the flow of melts, thus in this study, we used In-situ X-ray falling sphere viscometry experiments to determine the viscosity of the mantle incipient melt containing 1.8–4.0 wt.% water at 1.5–6 GPa and 2100 K. We found that water and pressure can effectively decrease the viscosity of the mantle incipient melt. Therefore, the mantle incipient melts are highly mobile with a high segregation velocity in the deeper part of the asthenosphere. However, during ascent, their mobility diminishes due to an increase in viscosity and a decrease in water content. In conclusion, these mechanisms potentially contribute to the accumulation of partial melt at the top of the asthenosphere.

58 GEOSCIENCES↗

Rheology and engine performance of very low sulfur fuel oil blended with 10% fast pyrolysis and hydrothermal liquefaction oils in a 2-stroke crosshead engine

The performance and emissions for a downscaled single-cylinder 2-stroke crosshead engine were determined for a very low sulfur fuel oil (VLSFO) when blended with 10 wt.% fast pyrolysis (FP) or hydrothermal liquefaction (HTL) bio-intermediates. The FP and HTL oils were derived from biomass and were observed to contain lower molecular weight (MW) hydrocarbons than neat VLSFO (which was evaluated as a baseline comparison). The addition of either biofuel reduced the overall viscosity of the VLSFO. Aging tests at 50, 90, and 120°C showed that the dynamic viscosity of VLSFO increased with exposure time up to two weeks. Similar trends were observed for the FP and HTL blends, but a pronounced spike in viscosity occurred for these fuels during the early period of exposure. None of the viscosity increases exceeded the operational limits of fuel system pumps. Engine performance studies were conducted under low, medium and high load operational settings. The relative performance of the test fuels was highly dependent on operating condition. In general, the engine results for the three test fuels were similar, but modest improvements in brake thermal efficiency and brake specific fuel consumption were observed, which may be attributed to the heightened reactivity of low molecular weight fraction of the FP and HTL oils.

09 BIOMASS FUELS↗

Computing the viscous effect in early-time drop impact dynamics

The impact of a liquid drop on a solid surface involves many intertwined physical effects, and is influenced by drop velocity, surface tension, ambient pressure and liquid viscosity, among others. Experiments by Kolinski et al. ( Phys. Rev. Lett. , vol. 112, no. 13, 2014 b , p. 134501) show that the liquid–air interface begins to deviate away from the solid surface even before contact. They found that the lift-off of the interface starts at a critical time that scales with the square root of the kinematic viscosity of the liquid. To understand this, we study the approach of a liquid drop towards a solid surface in the presence of an intervening gas layer. We take a numerical approach to solve the Navier–Stokes equations for the liquid, coupled to the compressible lubrication equations for the gas, in two dimensions. With this approach, we recover the experimentally captured early time effect of liquid viscosity on the drop impact, but our results show that lift-off time and liquid kinematic viscosity have a more complex dependence than the square-root scaling relationship. We also predict the effect of interfacial tension at the liquid–gas interface on the drop impact, showing that it mediates the lift-off behaviour.

42 ENGINEERING↗

Optimal white-noise stochastic forcing for linear models of turbulent channel flow

In the present study an optimisation problem is formulated to determine the forcing of an eddy-viscosity-based linearised Navier–Stokes model in channel flow at $Re_\tau \approx 5200$ ( $Re_\tau$ is the friction Reynolds number), where the forcing is white-in-time and spatially decorrelated. The objective functional is prescribed such that the forcing drives a response to best match a set of velocity spectra from direct numerical simulation (DNS), as well as remaining sufficiently smooth. Strong quantitative agreement is obtained between the velocity spectra from the linear model with optimal forcing and from DNS, but only qualitative agreement between the Reynolds shear stress co-spectra from the model and DNS. The forcing spectra exhibit a level of self-similarity, associated with the primary peak in the velocity spectra, but they also reveal a non-negligible amount of energy spent in phenomenologically mimicking the non-self-similar part of the velocity spectra associated with energy cascade. By exploiting linearity, the effect of the individual forcing components is assessed and the contributions from the Orr mechanism and the lift-up effect are also identified. Finally, the effect of the strength of the eddy viscosity on the optimisation performance is investigated. The inclusion of the eddy viscosity diffusion operator is shown to be essential in modelling of the near-wall features, while still allowing the forcing of the self-similar primary peak. In particular, reducing the strength of the eddy viscosity results in a considerable increase in the near-wall forcing of wall-parallel components.

Mechanics↗

Fuel Properties of Oxymethylene Ethers with Terminating Groups from Methyl to Butyl

Oxymethylene ethers (OMEs) have been studied as possible additives or replacements for diesel fuels. Typically, studies have considered only methyl-terminated OMEs. Recent structure-property relationship models suggest that extended-alkyl OMEs may provide improvements to many of the properties of methyl-terminated OMEs that make them less suitable as diesel fuel blendstocks. In this work, we describe the synthesis and characterization of 16 different OMEs with methyl, ethyl, propyl, butyl, isopropyl, and isobutyl terminating alkyl groups with varying oxymethylene chain length. Indicated Cetane Number, Lower Heating Value, Flash Point, Density, Viscosity, Vapor Pressure, and Oxidative Stability are tested via ASTM standard methods. Additionally, Water Solubility, Boiling Point, seal material compatibility, and sooting propensity (via the Yield Sooting Index) are measured for these fuels. For diesel compatibility, all tested OMEs except smaller methyl and ethyl OMEs, and the branched isopropyl OME, meet cetane number requirements. Further, extending the alkyl end group increases the heating value, but all OMEs, due to their oxygen content, have heating values less than diesel; despite this, all OMEs show significant reductions in soot production per unit heating value. Only the heaviest OMEs meet diesel viscosity requirements, and most are higher density than diesel. OMEs with larger alkyl groups show the highest stability under accelerated auto-oxidation conditions. Increases in alkyl group length cause order of magnitude reduction in water solubility, from hundreds of g/L for methyl terminated OMEs to hundreds of mg/L for butyl terminated OMEs. Limited seal material testing indicates that PEEK polymers are unaffected by OMEs; while extended alkyl groups may improve compatibility with FKM (Viton), other common elastomers (NBR, silicone) remain incompatible with all tested OMEs. Overall, it is found that methyl-terminated OMEs exhibit the most potential for soot reduction, but OMEs with larger propyl and butyl terminating alkyl groups show improved compatibility with existing diesel systems.

09 BIOMASS FUELS↗

Transport of Compressed Woody Biomass: Correlating Rheology and Microcompounder Measurements

Transport of milled solid biomass can be problematic in biomass conversion processes due to the material’s high cohesiveness, low bulk density, and poor flowability. Here, the goal is to investigate the flow behavior of compressed biomass under conditions similar to compression-screw feeders. To mimic the flow of woody biomass inside a compression-screw feeder, two lignocellulosic biomass feedstocks were fed to a co-rotating twin-screw microcompounder. Force and screw speed from the microcompounder were correlated to viscosity in a traditional rheometer, where low-density polyethylene served as a control. The deformation of compressed biomass in a lab-scale compounder is strongly dependent on screw speed and moisture content. Shear stress in woody biomass decreased with the shear rate, indicating negative plastic viscosity. Moreover, viscosity decreased one order of magnitude as screw speed increased from 10 to 90 rpm. Increasing moisture from 10 to 50% reduced the viscosity of the compressed biomass by 60%.

09 BIOMASS FUELS↗

Temperature Dependence of Volumetric and Dynamic Properties of Imidazolium-Based Ionic Liquids

Atomistically detailed molecular dynamics simulations were used to investigate the temperature dependence of the specific volume, dynamic properties, and viscosity of linear alkyl chain ([C n C 1 Im][NTf 2 ], n = 3–7) and branched alkyl chain ([(n – 2)mC n–1 C 1 Im][NTf 2 ]) ionic liquids (ILs). The trend of the glass transition temperature (T g ) values obtained in the simulations as a function of the alkyl chain length of cations was similar to the trend seen in experiments. In addition, the system relaxation behavior as determined from the temperature dependence of the diffusion coefficient, rotational relaxation time, and viscosity close to T g was observed to follow the Vogel–Fulcher–Tammann expression. Furthermore, the reciprocal of the diffusion coefficient of the anion and cation in both linear and branched IL systems showed a linear correlation with viscosity, thus confirming the validity of the Stokes–Einstein relationship for these systems. Similarly, the average rotational relaxation time of the ions was also found to correlate linearly with the viscosity of the ILs over a wide range of temperatures, thereby validating the Debye–Stokes–Einstein relationship for the ILs. These simulation findings suggest that the temperature dependence of the relaxation time of ILs is very similar to that of other glass-forming liquids.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Nonlinear Shear Rheology of Entangled Polymer Rings

Steady-state shear viscosity η($\bar{γ}$) of unconcatenated ring polymer melts as a function of the shear rate $\bar{γ}$ is studied by a combination of experiments, simulations, and theory. Experiments using polystyrenes with Z ≈ 5 and Z ≈ 11 entanglements indicate weaker shear thinning for rings compared to linear polymers exhibiting power law scaling of shear viscosity η ~ $\bar{γ}$ –0.56 ± 0.02 , independent of chain length, for Weissenberg numbers up to about 10 2 . Nonequilibrium molecular dynamics simulations using the bead-spring model reveal a similar behavior with η ~ $\bar{γ}$ -0.57 ± 0.08 for 4 ≤ Z ≤ 57. Viscosity decreases with chain length for high $\bar{γ}$. In our experiments, we see the onset of this regime, and in simulations, which we extended to Wi ~ 10 4 , the nonuniversality is fully developed. In addition to a naive scaling theory yielding for the universal regime η ~ $\bar{γ}$ –0.57 , we developed a novel shear slit model explaining many details of observed conformations and dynamics as well as the chain length-dependent behavior of viscosity at large $\bar{γ}$. The signature feature of the model is the presence of two distinct length scales: the size of tension blobs and much larger thickness of a shear slit in which rings are self-consistently confined in the velocity gradient direction and which is dictated by the size of a chain section with relaxation time 1/$\bar{γ}$. These two length scales control the two normal stress differences. In this model, the chain length-dependent onset of nonuniversal behavior is set by tension blobs becoming as small as about one Kuhn segment. This model explains the approximate applicability of the Cox–Merz rule for ring polymers.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Polymer Topology Effects on Dynamics of Comb Polymer Melts

Controlling polymer viscosity and flow is key to their many applications through strength and processability. The topology of the polymer i.e., linear, stars, and branched, affects the macroscopic flow characteristics of melts, where introducing one branch is sufficient to increase the viscosity significantly. While a number of studies have probed the effects of polymer topology on their rheology, the molecular understanding that underlies the macroscopic behavior remains an open question. The current study uses molecular dynamics simulations to resolve the effects of topology of polymer melts on chain mobility and viscosity in the comb regime using polyethylene as a model system. A coarse-grained model where four methylene groups constitute one bead is used, and the results are transposed to the atomistic level. Here, we find that while the number of branches only slightly affects the chain mobility and viscosity, their length strongly impacts their behavior. Finally, the results are discussed in terms of interplay between the relaxation of the branches and reptation of the backbone where the topology of the polymer affects the tube dimensions.

36 MATERIALS SCIENCE↗

Rheological Investigation on the Microstructure of Fuel Cell Catalyst Inks

We present a rheological investigation of fuel cell catalyst inks. The effects of ink parameters, which include carbon black-support structure, Pt presence on carbon support (Pt-carbon), and ionomer (Nafion) concentration, on the ink microstructure of catalyst inks were studied using rheometry in combination with ultrasmall-angle X-ray scattering (USAXS) and dynamic light scattering (DLS). Dispersions of a high-surface-area carbon (HSC), or Ketjen black type, demonstrated a higher viscosity than Vulcan XC-72 carbon due to both a higher internal porosity and a more agglomerated structure that increased the effective particle volume fraction of the inks. The presence of Pt catalyst on both the carbon supports reduced the viscosity through electrostatic stabilization. For carbon-only dispersions (without Pt), the addition of ionomer up to a critical concentration decreased the viscosity due to electrosteric stabilization of carbon agglomerates. However, with Pt-carbon dispersions, the addition of ionomer showed contrasting behavior between Vulcan and HSC supports. In the Pt-Vulcan dispersions, the effect of ionomer addition on the rheology was qualitatively similar to Vulcan dispersions without Pt. The Pt-HSC dispersions showed an increased viscosity with ionomer addition and a strong shear-thinning nature, indicating that Nafion likely flocculated the Pt-HSC aggregates. These results were verified using DLS and USAXS. Further, the observations of the effect of ionomer:carbon ratio and a comparison between carbons of different surface areas provided insights on the microstructure of the catalyst ink corresponding to the optimized I/C ratio for fuel cell performance reported in the literature.

36 MATERIALS SCIENCE↗

Reactive Modified Epoxy Resin and Its Miscible Blends Based on Recycled Oligomers from Solvolysis

Chemical depolymerization of fully cured epoxy resin with 20% reactive modifier was successfully performed via a solvent-assisted solvolysis process into low molecular weight recyclable oligomers (RO) at 240 °C in a pressure vessel at 650 psi for 4 h. The thermoset epoxy resin was depolymerized into transparent brown viscous fluid with a higher viscosity than the uncured epoxy resin with approximately 93% yield. Different concentrations of the RO were homogeneously mixed with the pure epoxy resin, and their curing kinetics, viscosity, FTIR, mechanical properties, DMA, and cross-link density were investigated. The curing kinetics of the pure reactive modified epoxy resin (baseline) and its mixtures with RO of different concentrations were investigated under both isothermal and nonisothermal conditions using small amplitude oscillatory shear flow. The elastic and viscous moduli (G′ and G″), complex viscosity (η*), and tan δ values were evaluated at different curing times and temperatures. The G′, G″, and η* increased dramatically, while tan δ decreased strongly by several orders of magnitude at the gel point. The zero-shear viscosity (η 0 ) was determined from the angular frequency dependent on η* based on the Cross model for different blend compositions in the liquid state before curing. The composition dependence of η 0 showed a positive deviation from the linear mixing rule and was well described by the Lecyar model. Here, the apparent activation energy of curing (E a ) was also evaluated according to the Arrhenius equation and was found to be 46 ± 2 kJ/mol regardless of the different contents of RO. For all blends up to 40 wt % RO, only one tan δ peak systematically shifting to lower temperatures with increasing content of RO was observed in the DMA measurements, indicating that the epoxy resin and the RO are miscible with up to 40 wt % RO.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗