Similarity, physical

Two systems are said to be physically similar in respect to certain specified physical quantities when the ratio of corresponding magnitudes of these quantities between the two systems is the same everywhere. 

Kinematic Lengths + time intervals, or velocities Planetarium, tidal models 

Dynamic Forces Flow models, wind tunnel 

Thermal Temperature differences Pilot-plant heat exchanger 

Chemical Concentration differences Bench scale reactor 

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Buckingham, E., 1914. On physically similar systems illustrations of the use of dimensional equations. Physics Review, 4, 335-376. 

The chemical and particularly the physical similarity between thiophene and benzene has intrigued chemists for a long time, and comparison between phenyl and thienyl derivatives has often been made. Accordingly, it seems somewhat surprising that the effect of replacement of a vinyl group, —CH=CH—, in benzene by has... 

Polyformaldehydes (polyoxymethylenes, polyacetals) These are physically similar to general purpose nylons but with greater stiffness and lower water absorption. There are no solvents, but swelling occurs in liquids of similar solubility parameter. Poor resistance to u.v. light and limited thermal stability are two disadvantages of these materials. 

Lord Ivelvin has pointed out that the similarity between the laws of gases and of dilute solutions carries with it no inference as to physical similarity between the states, although Boltzmann has developed a theory of osmotic phenomena which regards the pressure as due to a bombardment of the semipermeable wall by the molecules of dissolved solute, whilst it is subjected to equal and opposite forces by bombardment from the solvent molecules inside and outside. 

Massey, B. S. Units, Dimensional Analysis and Physical Similarity (van Nostrand Reinhold, 1971). 

Pure elemental silicon is a hard, dark gray solid with a metallic luster and with a crystalline structure the same as that of the diamond form of carbon. For this reason, silicon shows many chemical and physical similarities. There is also a brown, powdery form of silicon having a microcrystalline form. The element is prepared commercially by reducing the oxide by reacting it with carbon (as coke) in electric furnaces. On a small scale, silicon has been obtained from the oxide by reduction with aluminum meted. 

When the relationship between the material flux and the parameters of the system can be calculated directly by solution of the appropriate differential equations, the criterion equation (2.7.30) has little significance. However, this is not possible in the great majority of practical systems, and thus the empirically determined criterial equation is of general validity for physically similar systems. It can form a basis for designing larger equipment on the basis of experiments with model systems. 

Setting the block of frequency parameters can sometimes be omitted for subsequent similar samples if the spectrometer is sufficiently stable. Similar samples means that the sample tubes come from a single batch of quartz tubing (i.e., they have identical inner and outer diameter within a small tolerance) and the samples are physically similar (e.g., frozen buffered protein solutions). Such samples are invariant as dielectric, and they all will cause the same shift in resonator frequency within circa 1 MHz. 

The objects of statistical mechanics however, are not single systems, but ensembles of many, say v, physically similar systems, i.e. systems with Hamiltonians H(qi,pi,t). The distribution of such a virtual ensemble over the phase space will be described by means of the density function q(qi,Pi, t), called the phase density. It is a real non-negative quantity and will always be normalized to unity, i.e. 

The tracer should not be adsorbed on or react with the surface of the vessel. Alternatively, the tracer should be chemically and physically similar to the fluid flowing, so that any adsorption (or diffusion) behavior may be replicated. 

Bohr s theory not only solved the puzzle of the rare earths, it also explained why Mendeleev s periodic law works so well in most cases. Elements with the same number of electrons in the outermost shell have similar properties. For example, both sodium and potassium have a single electron in the outer shell. Magnesium and calcium, which are also chemically and physically similar, each have two. Carbon and silicon have four. And so on. 

Like the unsupervised methods, the supervised methods discussed in this book are based on the assumption that samples that are chemically or physically similar wM be near each other in measurement (row) space. 

The effect we will describe in this Section is physically similar to what in the past was called fluctuational barrier preparation , where proton transfer between two heavy atoms is facilitated by an oscillation that brings these heavy atoms closer so that it lowers the potential energy barrier (see Figure 1). 

Imagine that you have successfully scaled up a mixing or a granulating process from a 10-liter batch to a 75-liter and then to a 300-liter batch. What exactly happened You may say, I got lucky. Apart from luck, there had to be some physical similarity in the processing of the batches. Once you understand what makes these processes similar, you can eliminate many scale-up problems. 

An internal standard is a compound that is not present in the sample, but is chemically and physically similar to the analytes of interest. A fixed quantity is incorporated into the calibration solutions. The same concentration of internal standard is added to the samples during extraction to compensate for analyte recovery and injection variability. As seen in Figure F2.3.I, Echinenone, which is not typically found in foods, is used as the internal standard. Unfortunately, compounds which may be used as internal standards for carotenoid analysis are not readily available commercially. 

In an unpublished paper by Skalov (Institute of Chemical Physics) similar considerations were applied earlier to the theory of propagation of an auto-catalytic reaction by diffusion of the active product, without accounting for the reaction heat. 

The solubilities of various metals in mercury at 18° C. are noted in the accompanying table. Concentration of the saturated amalgam usually results in the precipitation of the dissolved solute as a mercuride, rarely in the form of the free metal. It should be noted that metals chemically and physically similar to mercury are characterized by high solubility. The farther removed in the Mendeleev periodic table the metals are from mercury the less soluble they become. Actually none of the metals except thallium is extremely soluble consequently, methods leading to the preparation of amalgams often give products... 

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