Log plots

Figure C2.5.3. Histogram of the number of stmctures with a given number of associated sequences for the 3D 3 X 3 X 3 case, in a log-log plot.

It should be noted that a log-log plot condenses the data considerably and that the transition between a first-power and a 3.4-power dependence occurs over a modest range rather than at a precise cutoff. Nevertheless, the transition is read from the intersection of two lines and is identified as occurring at a degree of polymerization or molecular weight designated n, or, respectively. 

Figure 3.9 Log-log plots of modulus versus time for polyisobutylene at 25 C and polystyrene at 135°C. Note the different units of time for the two substances. (From data of A. V. Tobolsky and E. Catsiff and of H. Fujita and K. Ninomiya. From Ref. 4.)...

Figure 3.12 Log-log plots of compliance versus time for polystyrene at 100 C and cis-polyisoprene at -30°C. (Data of D. J. Plazek and V. M. O Rourke and of N. Nemoto, M. Moriwaki, H. Odani, and M. Kurata from Ref. 4.)...

Figure 4.9 Log-log plot of ln(l - 6) versus time for poly(ethylene tereph-thalate) at three different temperatures. [Reprinted from L. B. Morgan, Philos. Trans. R. Soc. London 247A 13 (1954).]...

Figure 6.3 Log-log plots of Rp versus concentration which verify the order of the kinetics with respect to the constituent varied, (a) Monomer (methyl methacrylate) concentration varied at constant initiator concentration. [Data from T. Sugimura and Y. Minoura, J. Polym. Sci. A-l 2735 (1966).] (b) Initiator concentration varied AIBN in methy methacrylate (o), benzoyl peroxide in styrene ( ), and benzoyl peroxide in methyl methacrylate ( ). (From P. J. Flory, Principles of Polymer Chemistry, copyright 1953 by Cornell University, used with permission.)...

Figure 9.8 Log-log plot of [7 ]q versus M for four different polymer-solvent-temperature combinations corresponding to 0 conditions. All lines have a slope of 1/2 as required by Eq. (9.54). (Reprinted with permission from Ref. 1.)...

Prepare a log-log plot of rx versus X and evaluate the slope as a test of the Rayleigh theory applied to air. The factor M/pN in Eq. (10.36) becomes 6.55 X 10 /No, where Nq is the number of gas molecules per cubic centimeter at STP and the numerical factor is the thickness of the atmosphere corrected to STP conditions. Use a selection of the above data to determine several estimates of Nq, and from the average, calculate Avogadro s number. The average value of n - 1 is 2.97 X 10" over the range of wavelengths which are most useful for the evaluation of N. ... 

Fig. 10. Log—log plot scale-up by power per unit volume where for A, constant blend time, = 2/3 B, same vortex, y = 1/6 C, dispersion, = 0 D,...

According to the Scher-MontroU model, the dispersive current transient (Fig. 5b) can be analyzed in a double-log plot of log(i) vs log(/). The slope should be —(1 — ct) for t < and —(1 + a) for t > with a sum of the two slopes equal to 2, as shown in Figure 5c. For many years the Scher-MontroU model has been the standard model to use in analyzing dispersive charge transport in polymers. 

Compressibility. The bulk density of a soHd is an essential value used in the analysis of its flow properties, such as when calculating mass flow hopper angles, opening sizes, bin loads, etc. Loose and/or packed density values ate not sufficient. Bulk soHds exhibit a range of densities that vary as a function of consoHdating pressure. This range of densities, called the compressibiHty of the soHd, can often be expressed on a log—log plot as a line or relationship. 

The significance of G G tan 5, Tj, and Tj is that they can be determined experimentally and used to characterize real materials. These parameters depend on frequency and temperature, and this dependence can be used to define behavior. For example, viscoelastic fluids are often characterized by log—log plots of one or more of these quantities vs the angular frequency CO, as shown in Figure 21, which illustrates the behavior of a polymer melt (149). 

Polymer melts are frequendy non-Newtonian. In this case the earlier expression given for the shear rate at the capillary wall does not hold. A correction factor (3n + 1)/4n, called the Rabinowitsch correction, must be appHed in such a way that equation 21 appHes, where 7 is the tme shear rate at the wall and nis 2l power law factor (eq. 22) determined from the slope of a log—log plot of the tme shear stress at the wad, T, vs 7. For a Newtonian hquid, n = 1. A tme apparent viscosity, Tj, can be calculated from equation 23. 

Fig. 12. Log—log plot of current density,/, versus appHed electric field, E, for a ZnO varistor at room temperature, ia which the breakdown field. Eg, is iadicated. The exponent d equals the iaverse slope of the curve, log Ej J) = 1/a, and is a measure of device nonlinearity. Units of current density and the...

Fig. 23. Log—log plot of pressure drop per unit height of typical packing as a function of gas rate at two Hquid rates and for the unirrigated packing.

Published Cost Correla.tions. Purchased cost of an equipment item, ie, fob at seller s site or other base point, is correlated as a function of one or more equipment—size parameters. A size parameter is some elementary measure of the size or capacity, such as the heat-transfer area for a heat exchanger (see HeaT-EXCHANGETECHNOLOGy). Historically the cost—size correlations were graphical log—log plots, but the use of arbitrary equation forms for correlation has become quite common. If cost—size equations are used in computer databases, some limit logic must be included so that the equation is not used outside of the appHcable size range. 

One simple equation form is the exponential equation, which gives a straight line on a log—log plot ... 

When a distribufion of particle sizes which must be collected is present, the aclual size distribution must be converted to a mass distribution by aerodynamic size. Frequently the distribution can be represented or approximated by a log-normal distribution (a straight line on a log-log plot of cumulative mass percent of particles versus diameter) wmich can be characterized by the mass median particle diameter dp5o and the standard statistical deviation of particles from the median [Pg.1428]

Fig. 1. Log-log plot of the contact radius as a function of particle radius for soda-lime glass particles on polyurethane (from ref. [56]).

Induction period measurements can also be used to determine interfacial tensions. To validate the values inferred, however, it is necessary to compare the results with an independent source. Hurley etal. (1995) achieved this for Cyanazine using a dynamic contact angle analyser (Calm DCA312). Solid-liquid interfacial tensions estimated from contact angle measurements were in the range 5-12 mJ/m which showed closest agreement with values (4—20mJ/m ) obtained from the log-log plots of induction time versus supersaturation based on the assumption of — tg. 

FIG. 7 Log-log plots of the interface width (w ) versus the Monte Carlo time t, measured at different adsorption probabihties using channels of width L = 30. Data were obtained during the displacement of an A-poisoned phase by the reactive regime. From top to bottom the probabihties are 0.5192, 0.5202, 0.5211, 0.5215, and 0.5238. 

FIG. 14 Semi-log plot of mean chain length L vs width of the open slit D at various temperatures in 3d. Full symbols denote flexible chains and empty symbols semirigid chains with activation energy a = 0.5 [61]. 

FIG. 8 Log-log plot of the relaxation time r23, Eq. (25), vs chin length N, for four values of e. Full straight lines indicate power-law fits including the shortest chain length N — 6 the broken line indicates a fit where A = 16 is excluded. Effective exponents Zgff are quoted [13],... 

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