The Similarity Principle

Physical modeling involves searching for the same or nearly the same similarity criteria for the model and the real process. The full-scale process is modeled on an increasing scale with the principal linear dimensions scaled-up in proportion, based on the similarity principle. For relatively simple systems, the similarity criteria and physical modeling are acceptable because the number of criteria involved is limited. For complex systems and processes involving a complex system of equations, a large set of similarity criteria is required, which are not simultaneously compatible and, as a consequence, cannot be realized. 

The next step toward the similarity principles is to develop the governing equations in a nondimensional form. The equations are normalized by first defining the dimensionless, independent variables as... 

It is difficult to obtain the correct temperature boundary conditions in a model. Radiation between surfaces in a room and conduction throu the surfaces are important for the level of the surface temperature T, x,y,z). It is difficult to establish the similarity principles based on radiation and conduction. A practical method is to estimate the influence of radiation and conduction and include this level in the boundary values of the model. In this way it... 

This section describes how this set of equations can be solved analytically by the introduction of various simplifications. First, gas dynamics is linearized, thus permitting an acoustic approach. Next, a class of solutions based on the similarity principle is presented. The simplest and most tractable results are obtained from the most extensive simplifications. 

Nature is sometimes difficult but never insidious. Fortunately, the similarity principle—compounds with similar chemical structures often possess similar properties or activities—is valid in many cases, and thus gives multivariate models a chance. 

The similarity principle allows one to formulate answers to questions as to how to construct a representative sample of a large combinatorial library, or how to design an efficient library to follow up an active lead compound. We can now postulate that a representative sample that contains molecules that are too similar will be wasteful, as the active structure will probably be duplicated. However, for lead follow-up we require the molecules to be... 

Equation 8.2 shows how the net flux density of substance depends on its diffusion coefficient, Dj, and on the difference in its concentration, Ac] 1, across a distance Sbl of the air. The net flux density Jj is toward regions of lower Cj, which requires the negative sign associated with the concentration gradient and otherwise is incorporated into the definition of Acyin Equation 8.2. We will specifically consider the diffusion of water vapor and C02 toward lower concentrations in this chapter. Also, we will assume that the same boundary layer thickness (Sbl) derived for heat transfer (Eqs. 7.10-7.16) applies for mass transfer, an example of the similarity principle. Outside Sbl is a region of air turbulence, where we will assume that the concentrations of gases are the same as in the bulk atmosphere (an assumption that we will remove in Chapter 9, Section 9.IB). Equation 8.2 indicates that Jj equals Acbl multiplied by a conductance, gbl, or divided by a resistance, rbl. 

Because an air packet and the molecules within it move as a unit, the eddy diffusion coefficients for different gaseous species are equal. In fact, Kj is often assumed to be the same for the transfer of gases, heat, and momentum (expressed in the same units), a relation that is referred to as the similarity principle. Therefore Kj is generally measured for the most... 

A patent for desulfurization of petroleum streams containing condensed ring heterocyclic organosulfur compounds on the basis of the similar principle was filed by Research and Engineering Company in 2001.58... 

Since authentic carbonyl diiodide has never been prepared, no data exist concerning the measurement of its physical properties. However, the boiling temperature of COIj has been estimated as 191 C by Horvath [985], based on the similarity principle (see Fig. 15.1), and taking into account its molecular mass of 281.82 g mol". ... 

The boiling temperature of COCII has been estimated at 69 C from the similarity principle, taking into account the relative molecular mass of the material at 190.37 (see Fig. 15.1) [985]. The critical properties for COCII, (542.0 K), (6.08 MPa) and... 

Here, the microfluidic actuation principles that are utiziled in massively parallel analysis are outlined briefly. This is followed by some commercial application examples. Due to the similar principle, microarrays and picowell plates are presented together, followed by bead based assays. 

The similar principle can be constructed also for the physiological secludedness (see Beldeinishev, 1964). 

Applying the similar principles, we could cmitrol the fluid into multiple microchannels based on the needs. For such multiple-output modes, there are three different operation methods, as shown in Fig. 9, to control the fluid into the microchannels 2 and 4, for example, in the capillary system with a 1 X 4 microfluidic switch. The time-sequence actuation as shown in Fig. 9a, we would generate the growth and collapse of the thermal bubble six times to cmitrol the fluid to pass through the microchannels 4 and 2. The bubble pressure overcomes the barrier pressure to make the fluid pass through all hydrophobic patches of microchannels 2 and 4. Then, the capillary force pulls the fluid through to turn on microchannels 2 and 4. However, Fig. 9b demonstrates the same switch functimi with only... 

Using the similar principle of contact imaging, an optofluidic fluorescence microscope was fabricated [6]. Here an array of Fresnel zone plates (FZP) were used to generate focused light spots inside the chip. Similar to the OFM, here the FZP is fabricated in a diagonal fashion so that all the portions of the particle can be captured by the sensor (Fig. 2b). Unlike OFM, apertures are not created over the CMOS sensor, but instead specific filters are coated over the CMOS sensor to assist in the detection of fluorescence (Fig. 2a). The performance of the microscope seemed good when compared to... 

The majority of SAR analyses in today s medicinal chemistry projects are based on the similarity principle, as formulated by Johnson and Maggiora in 1990 [7]. It states that chemically similar molecules are likely to exhibit similar physicochemical and biological properties. Of course, this fundamental assumption was already explored in medicinal chemistry decades earlier and even before the advent of computational tools, resulting in the discovery of many clinical development candidates. 

Experience in medicinal chemistry has taught that there are significant exceptions to the similarity principle in many series, hi fact, oflai small changes in a lead slmcture completely destroy biological activity or— in a more favorable manner— result in a boost of potency for a target. This well-known phenomenon has recently been coined activity chfF [6] or activity hotspot [5] we will therefore use both terms in our discussion. 

Many instructive exceptions of the similarity principle were summarized by Kubinyi [48,49]. S ome of these differences could be related to unexpected 3D-bmding modes of ligands in the protein cavity after only minor chemical modifications. These examples show that it is very difficult to describe chemical similarity in an objective and global manner without considering the protein environment. In fact, lead optimization often takes advantage on these surprises and further explores such activity cliffs for biological activity. Regions around early hit structures with a flat SAR are typically less often explored. 

All rescaffolding approaches are built on two fundamental concepts The first one is the similarity principle, the other one is the concept of bioisosterism [123-125], which was formally introduced by Friedmann [126], based on work by Langmuir [127] and Erlenmeyer [128]. Here, those compounds are regarded as bioisosteres that show some structural differences but still share similar biological activity. Later this concept was employed for the replacement of functional groups with the aim to improve ADMET properties of compounds [129,130]. 

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