A Expansion factor

Figure 5. (a) Expansion factor of mean-square end-to-end distance aj and fbj radius of gyration as function of z = xBJN for open (Op.) and periodic (Per.) chain. Other results shown here are Domb and Barrett (DB) [67], Douglas and Freed (DF) [66], Muthukumar and Nickel (MN) [64], and Flory modified (Fm) [17,69]. (From ref. 68, by permission of the publishers, Butterworth Co. (Publishers) Ltd. .)... 

Figure 4.16 Dependence of (a) expansion factor i and (b) the effective degree of ionization on monovalent salt concentration for N=1000, b = 3, w = 0, and 3 = 3.5. (From Muthukumar, M., J. Cbem. Pbys., 120, 9343, 2004. With permission.)...

The relationship required is the gas expansion factor (E), and is defined for a given quantity (mass or number of moles) of gas as... 

In a good solvent, the end-to-end distance is greater than the 1q value owing to the coil expansion resulting from solvent imbibed into the domain of the polymer. The effect is quantitatively expressed in terms of an expansion factor a defined by the relationship... 

We shall defer a quantitative discussion of this expansion factor until the discussion of solutions in Chaps. 8 and 9. 

We saw in Sec. 1.11 that coil dimensions are affected by interactions between chain segments and solvent. Both the coil expansion factor a defined by Eq. (1.63) and the interaction parameter x are pertinent to describing this situation. 

Conditions in which the effects of item (2) exactly compensate are called 0 conditions. The expansion factor a gives the ratio of coil dimensions under non-0 conditions to those under 0 conditions. 

Next we consider the situation of a coil which is unperturbed in the hydro-dynamic sense of being effectively nondraining, yet having dimensions which are perturbed away from those under 0 conditions. As far as the hydrodynamics are concerned, a polymer coil can be expanded above its random flight dimensions and still be nondraining. In this case, what is needed is to correct the coil dimension parameters by multiplying with the coil expansion factor a, defined by Eq. (1.63). Under non-0 conditions (no subscript), = a(rg)Q therefore under these conditions we write... 

Next we shall examine the molecular weight dependence of the coil expansion factor a to see if the latter can explain the observations of a s greater than 0.5. 

Our primary objective in undertaking this examination of the coil expansion factor was to see whether the molecular weight dependence of a could account for the fact that the Mark-Houwink a coefficient is generally greater than 0.5 for T 0. More precisely, it is generally observed that 0.5 < a < 0.8. This objective is met by combining Eqs. (9.55) and (9.68) ... 

What is especially significant about Eq. (9.68) is the observation that the coil expansion factor a definitely increases with M for good solvents, meaning that-all other things being equal longer polymer chains expand above their 0 dimensions more than shorter chains. Even though the dependence of a on... 

Rate of discbarge through an orifice meter is given by Eq. (10-8) for either hquids or gases. For the case of subsonic flow of a gas (/ < / < 1.0), the expansion factor Y for orifices is approximated by... 

Part AM This part lists permitted individual constnic tion materials, apphcable specifications, special requirements, design stress-intensity vafues, and other property information. Of particular importance are the ultrasonic-test and tou ness requirements. Among the properties for which data are included are thermal conduc tivity and diffusivity, coefficient of theiTnal expansion, modulus of elasticity, and yield strength. The design stress-intensity values include a safety factor of 3 on ultimate strength at temperature or 1.5 on yield strength at temperature. 

Double Layer Pipe expansion is a significant factor at tenmera-tures above 600°F (3I6°C). Above this temperature, insiilation should be apphed in a double layer with all joints staggered to prevent excessive heat loss and high surface temperature at joints opened by pipe expansion. This procedure also minimizes thermal stresses in the insulation. 

Expanders have not been the essence of reliability. It is not that the expander design in itself has any significant problems. The problems for the most part seem to be related to the application. Most of the failures have been the result of the expander ingesting foreign substances, such as the catalyst in a catalytic cracking unit heat recovery application. Unlike the expansion section of the gas turbine, the inlet temperature is not as high, therefore, temperature is not a significant factor in reliability reduction. 

A key factor in the suitabihty of cokes for graphite production is their isotropy as determined by the coefficient of thermal expansion. After the calcined coke was manufactured into graphite, the axial CTE values of the graphite test bars were determined using a capacitance bridge method over a temperature range of 25 to 100°C. The results are summarized in Table 24. Also included in the table are bulk density measurement of calcined cokes and the resistivity values of their graphites. 

For the discharge of compressible fluids from the end of a short aiping length into a larger cross section, such as a larger pipe, vessel, or atmosphere, the flow is considered adiabatic. Corrections are applied to the Darcy equation to compensate for fluid property changes due to the expansion of the fluid, and these are known as Y net expansion factors [3]. The corrected Darcy equation is ... 

Figure 2-38A. Net expansion factor, Y, for compressible flow through pipe to a larger flow area. By permission, Crane Co., Technical Paper U410, Engineering Div., 1957. Also see 1976 edition.

The function g(r j) is here a correlation factor, and it is clear that, if this function is chosen in different ways, we will have different coefficients CK in the expansion in the first factor. An important problem is then to try to find forms of the correlation factor which will reduce this expansion as much as possible, and a particular problem is to investigate how good an approximation could be obtained by a single determinant ... 

The convergency of expansion IV.3 may also be improved by introducing a correlation factor g ... 

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