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Motion in a circle
Published in John Bird, Carl Ross, Mechanical Engineering Principles, 2019
Let r = radius of horizontal turning circle,L = length of string,h = OC,ω = constant angular velocity about C,m = mass of particle P,T = tension in string, and θ = cone angle
Introduction to Organometallics
Published in Samir H. Chikkali, Metal-Catalyzed Polymerization, 2017
Samir H. Chikkali, Sandeep Netalkar
Ligands play a central role in organometallic chemistry and subsequent catalysis. Therefore, any ligand designing needs to be based on certain rationales. To assist rational ligand designing, various general concepts have been developed in the recent past. In his efforts to quantify ligand properties, Tolman introduced the concept of cone angle (Θ).7 For monodentate ligands, this angle is considered as ligand cone angle or Tolman cone angle that can be extended to any monodentate ligand. It is simply the measure of the steric size of ligand. If we consider that the monodentate phosphine ligand (such as trimethyl phosphine) is coordinated to the metal at the axial position, the cone angle may be defined as an angle subtended by the cone with the metal at the vertex and the hydrogen (or any other atom for other ligands) at the perimeter of the cone (Figure 1.10).
Introduction
Published in Jubaraj Bikash Baruah, Principles and Advances in Supramolecular Catalysis, 2019
Efficient and selective catalysts are developed by introducing supramolecular features to a conventional catalyst. For example, 1-octene reacts with carbon monoxide and hydrogen to undergo hydroformylation reaction as well as an isomerization reaction in the presence of catalytic amount of the complex [Rh(H)(PR3)2(CO)2] (R═aryl, alkyl) to yield nonan-1-al, nonan-2-al and 2-octene. The [Rh(H)(PR3)2(CO)2] complex adopts two different isomeric structures in solution. A similar complex to this rhodium complex but with a different composition, [Rh(H)(CO)3(PR3)] shows catalytic reactivity for the above reaction differently. In terms of the rate of the formation of product, this compound has better catalytic activity. The reaction is nonselective for the formation of nonan-1-al. The catalytic hydroformylation reaction by the catalyst [Rh(H)(PR3)2(CO)2] is a relatively slow reaction but provides nonan-1-al as a major product over the others. Based on such observations, new catalysts for hydroformylation reactions are developed by introducing a pyridine-based phosphine-capped ligand 1.15b. This ligand has nitrogen atom/s of pyridine/s anchored to zinc-porphyrin complexes to make highly hydrophobic environments. The ligand has also a large cone angle; hence, limited amounts of Zntpp can go into the coordination sphere of palladium. Thus, only one Zntpp ligand (Figure 1.15) binds to the palladium ion. Accordingly, the phosphorus site of tpp and Zntpp independently ligates to rhodium (I) ion and in-situ complexes [Rh(acac)(tpp)(CO)] and [Rh(acac)(Zntpp)(CO)] (where acac═acetylacetonate anion), respectively, are generated for the catalytic hydroformylation reaction. Both the complexes are active catalysts for the hydroformylation reaction of 1-octene at 80°C and 20 bar pressure. The rhodium complexes[Rh(acac)(tpp)(CO)] and [Rh(acac)(Zntpp)(CO)] drastically influence the catalytic hydroformylation process. With the catalytic amounts of these ligands, together with the [Rh(CO)2(acac)] catalyst, the catalytic turnover is doubled compared to using tpp than by using the ligand Zntpp. The percentage ratio of isomers relative to each other is also nearly doubled when Zntpp is used. The ratio of linear to branched aldehyde formed in the case of a catalytic reaction using tpp is 2.8, and for Zntpp it is 1.5. Thus, the extent of supramolecular features through hydrophobic sites and steric factors guides the ratios of linear and branched chain products as well as the enantiomers.
Patient positioning by visualising surgical robot rotational workspace in augmented reality
Published in Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2021
Marek Żelechowski, Murali Karnam, Balázs Faludi, Nicolas Gerig, Georg Rauter, Philippe C. Cattin
For surgeries that involve RCM, the range of angles that the endoscope can take while keeping the insertion port fixed is essential. After each trial, we logged two metrics: the surface coverage in percentages and the cone angle in degrees. The two metrics chosen represent the range of motion of the endoscope possible at the selected workspace map location. The surface area represents the total orientation change in either roll and yaw directions. In contrast, the cone angle represents the maximal range of motion possible in any direction of roll-yaw combined while the position is fixed. The surface coverage was calculated as a percentage of the area of the sphere reachable by the robot at a given position in the workspace. The cone angle expressed in degrees was chosen to provide a more intuitive way of comparing solid angles usually expressed in steradians (see Figure 4). We calculated the cone angle as the apex angle of the resulting cone, with a value range between 0 and 180°.
Advanced Gravity Concentration of Fine Particles: A Review
Published in Mineral Processing and Extractive Metallurgy Review, 2018
They observed that best separation efficiency was achieved when the cone angle was between 45° and 90°. At a very low cone angle of 10°, the WOC acted like a size classifier sending all the magnetite to the underflow and rejecting only 5% of the quartz. On the contrary, with 180° cone angle, only 15% magnetite reported to the underflow with 90% rejection of quartz. A longer vortex finder led to more solids reporting to the overflow accompanied by a decrease in magnetite recovery and an increase in quartz rejection. The separation efficiency dropped slightly with an increase in the vortex finder length initially and then dropped significantly.
Evaluation of the effect of cone geometry on spouted bed fluid dynamics by CFD-DEM simulation
Published in Drying Technology, 2022
An inverse behavior was observed between cone angles of 45° and 60°, with higher solids mass flow rate found for the cone angle of 60°. This could have been due to the larger diameter of the spout channel surface and the fact that the angle of the converging cone of the Venturi air distributor also had an angle of 60°, which may have favored solids circulation and, consequently, increased solids mass flow rate for the cone angle of 60°.