Weighting exponent

What weight of installed stainless-steel tank could have been obtained for the same capital investment as in the previous problem The 1980 cost for an installed 304 stainless-steel tank weighing 300,000 lb was 670,000. The installed cost-weight exponent for stainless tanks is 0.88 for a size range from 300,000 to 700,000 lb. 

Here. .. depends on the W-kernels introduced above, but in general stands for terms of the form cross-section, appropriately weighted by factors [1 — cosl(0)], l = 1, 2,. .., if the efficiency of exchange of quantity A in a collision depends on the scattering angle 0 in this way (as, e.g., in case of elastic neutral particle - ion collisions, see Sect. 2.2.4). In case of inelastic collision processes a is simply the total cross-section, denoted detector functions qa must be obtained by numerical integration and tabulated for the parameters of the relevant distribution functions fa. 

Fig. 3.5 Allometric relationship between unbound antipyrine intrinsic clearance (C/U(int)) per maximum life-span potential (MLP) and body weight. Note that the body weight exponent of the regression equation is close to unity. Units on the ordinate (litres per MLP) are equivalent to C/U(int) (L/min) x MLP (min). (Reproduced with permission from Boxenbaum (1982).)...

Chemical symbol for generic collision partner Initial mean molecular weight Exponent describing temperature dependence of rate coefficient kf ... 

Narh KA, Odell JA, Keller A (1992) Temperature-dependence of the conformational relaxation-time of polymer-molecules in elongational flow-invariance of the molecular-weight exponent. J Polym Sci Polym Phys 30 335-340... 

However, experiments (25) have indicated that the molecular weight exponent is close to 3.4. From the mc els that have been proposed to improve the reptation approach, it appears that in concentrated polymer solutions in good solvents, the solution viscosity is given as... 

In any case, whatever the model, since tube-renewal is more important (compared to reptation) and accelerates the motion more efficiently for short chains than for long chains, it introduces additional molecular weight dependences of the dynamical quantities, and certainly contributes to the experimental deviation of the viscosity/molecular weight exponent from the reptation value 3. All treatments, including tube renewal, exhibit such deviations which vanish for asymptotically long chains. Detailed quantitative tests are, however, very difficult to perform when tube-renewal is taken into account, polydispersity becomes an essential parameter (the shortest mechanism, reptation and tube-renewal, dominates the relaxation process). No complete set of experiments, either for diffusion or for viscoelasticity, with constant polydispersity at all molecular weights, are presently available. 

To summarize its applications, Eq. 8.2 describes most sets of measurements accurately. Root-mean-square fractional errors of 6-18% are found. The molecular weight dependence of the small-concentration diffusion coefficient is determined by the exponent a, which is consistently —0.5. The concentration exponent v is in the range 0.5-0.75, except for Brown, et al. s measurements, which lead to V 0.93(2). The molecular weight exponent y is modestly less than 0.5, namely between 0.32 and 0.46, except that Brown, et al. s measurements require y 0.6. The joint stretched exponential in c and M provides a good description for each set of measurements, with one consistent systematic deviation the approximation of using the same v for all polymer molecular weights is imperfect. 

In Ref. 118 (Fig. 8), the intrinsic viscosity [g] is compared with the weight average molecular weight. It was found that the absolute value of [ ] and the slope of the curve (on the log-log scale) are much lower for solutions of branched than of linear polymers. This experimental result, together with the percolation expressions for the intrinsic viscosity (Eqs. (21, 22)) confirm that clusters undergo hydrodynamic interactions. In fact, the molecular weight exponent value of [ly] is much lower with than without hydrodynamic interactions. This result implies that a calculation of the viscosity of the reaction bath is correct only if hydrodynamic interactions are taken into account. 

---