End corrections

Note that as M the absolute number of chain ends per unit volume decreases, as does the chain-end correction. 

In this case, no end corrections are apphed even though the formula apphes only for sharp-edged weirs. See Schoder and Dawson, Hydraulics, McGraw-Hill, New York, 1934, p. 175, for further details. 

Saa = -esB = e was chosen, and an effective coordination number Zcir = 2.5 occurred ) Thus one finds that the coefficient in the relation T N ,r is somewhat smaller than expected from Eq. (3), and in addition there is a "chain end"-correction. However, an integral equation theoryyielded Tc cc Vn, and th simulations showed that this result clearly is incorrect. More recent improved versions of integral equation theories now yield Tc N as well but still the theoretical understanding of the prefactor in this relation is limited. 

The original Simpson s formula without end correction may be generalized in a similar way as the trapezoidal formula for n = 2 panels, using Ax = (b - a)/2 and increasing k until sufficient accuracy is achieved, where... 

The results in Table 9.2 apply when no end effects are present. Sutterby (S7) determined simultaneous wall and end correction factors for the creeping flow range. His correlations are shown in Fig. 9.3 where the cylindrical column has closed ends a distance apart and the center of the spherical particle is distance Z from one end of the tube. The curve for D/L = 1.0 and Z/L = 1/2 is... 

Similarly, one must worry about exit pressure effects, or end corrections. Experimentally, it has been observed that there is a nonzero gange pressnre at the capillary exit, Pex (cnrve b of Fignre 7.70). It has been found that the ratio of exit to entrance effects, Pex tAPent, is between 0.15 and 0.20 and that althongh APgnt does not depend on L/Do, Pex decreases up to L/Dq = 10 and then remains constant. With these observations in mind, we can rewrite Eq. (7.68) to inclnde both entrance and exit effects ... 

The mean square length acquires an expansion factor which is independent of n, and end-correction terms of order less than n, so that... 

F or L/D > 100, the error in neglecting the end correction by using the fully developed pipe flow equation (6-81) is less than 2 percent. For rectangular channels, see Normand (Ind. Eng. Chem., 40, 783-787 [1948]). 

Also worth noting in connection with the foregoing discussion is the experimental observation of a nonzero gauge pressure at the capillary exit PeK (Curve b of Fig. 12.5). Thus one must take into consideration end corrections, both entrance and exit, and not only entrance corrections alone, as was done by Bagley. Sakiades (6) was the first to report the existence of Pex for polymer solutions, and Han and his co-workers have studied it extensively using polymer melts (7-9). It has been found that Pex/APent is between 0.15 and 0.20, and that, although APent does not depend on L/Dq, Pex decreases up to L/Dq = 10, then remains constant (9). 

H. L. Weissberg, End Corrections for Slow Viscous Flow through Long Tubes, Phys. Fluids, 5, 1033 (1962). 

The elementary theory, still too often used in calculating the surface tension, states that the pull required to detach the object from the surface is equal to the total perimeter of the wire multiplied by the surface tension. For a thin vertical plate, or a thin horizontal wire, the perimeter would be twice the length. There are, however, end corrections of uncertain magnitude, and accurate results cannot be obtained by the use of this over-simplified theory. As far as the author s experience goes, if elementary theory must be used, Wilhelmy s vertical... 

Although the Isothermal gas chromatographic column did not measure the heavy ends correctly, the results obtained with It give good comparisons of the composition and quality of the alkylates on essentially a heavy-end-free basis. These qualities are reported as the research octane number (RON) that was calculated (2) based on the composition of the alkylates as determined analytically. On essentially a heavy-end-free basis, RON values of 98-101 were common. Reanalyses of the alkylate In order to determine the heavy ends more correctly Indicated that the above RON values were In general too high by 3-4 units. RON values of 94.5 to 98 are thought to have occurred. 

Bagley EB (1957) End corrections in the capillary flow of polyethylene. J Appl Phvs 28 624-7. 

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