Power dependence

As with the steam turbine, if there was no stack loss to the atmosphere (i.e., if Qloss was zero), then W heat would he turned into W shaftwork. The stack losses in Fig. 6.34 reduce the efficiency of conversion of heat to work. The overall efficiency of conversion of heat to power depends on the turbine exhaust profile, the pinch temperature, and the shape of the process grand composite. 

To account for a first-power dependence of viscosity on molecular weight for lower molecular weights. 

To account for a 3.4-power dependence at higher molecular weights. 

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. 

To the extent that the segmental friction factor f is independent of M, then Eq. (2.56) predicts a first-power dependence of viscosity on the molecular weight of the polymer in agreement with experiment. A more detailed analysis of f shows that segmental motion is easier in the neighborhood of a chain end because the wagging chain end tends to open up the structure of the melt and... 

Equation (2.56) not only enables us to understand the basis for the first-power dependence of rj on M, but also presents us with a new and important theoretical parameter, the segmental friction factor. We shall see in the next chapter that it is a quantity which can also be extracted from measurements of the viscoelasticity of polymers. 

Equation (2.61) predicts a 3.5-power dependence of viscosity on molecular weight, amazingly close to the observed 3.4-power dependence. In this respect the model is a success. Unfortunately, there are other mechanical properties of highly entangled molecules in which the agreement between the Bueche theory and experiment are less satisfactory. Since we have not established the basis for these other criteria, we shall not go into specific details. It is informative to recognize that Eq. (2.61) contains many of the same factors as Eq. (2.56), the Debye expression for viscosity, which we symbolize t . If we factor the Bueche expression so as to separate the Debye terms, we obtain... 

What power dependence on M does r display according to these results Comment on the significance of the 3.4-power law according to these data and the results of the last problem. 

Inspection of Fig. 3.9 suggests that for polyisobutylene at 25°C, Ti is about lO hr. Use Eq. (3.101) to estimate the viscosity of this polymer, remembering that M = 1.56 X 10. As a check on the value obtained, use the Debye viscosity equation, as modified here, to evaluate M., the threshold for entanglements, if it is known that f = 4.47 X 10 kg sec at this temperature. Both the Debye theory and the Rouse theory assume the absence of entanglements. As a semi-empirical correction, multiply f by (M/M. ) to account for entanglements. Since the Debye equation predicts a first-power dependence of r) on M, inclusion of this factor brings the total dependence of 77 on M to the 3.4 power as observed. 

Although two of the mechanisms presented above yield the same power dependence on t, it appears possible to eliminate certain mechanisms by experimentally testing the development of 9 with time. A strategy for this is suggested by Eq. (4.28). Taking the logarithm of both sides of that equation gives... 

This result is called the Poiseuille equation, after Poiseuille, who discovered this fourth-power dependence of flow rate on radius in 1844. The poise unit of viscosity is also named after this researcher. The following example illustrates the use of the Poiseuille equation in the area where it was first applied. 

The nuclear chain reaction can be modeled mathematically by considering the probable fates of a typical fast neutron released in the system. This neutron may make one or more coUisions, which result in scattering or absorption, either in fuel or nonfuel materials. If the neutron is absorbed in fuel and fission occurs, new neutrons are produced. A neutron may also escape from the core in free flight, a process called leakage. The state of the reactor can be defined by the multiplication factor, k, the net number of neutrons produced in one cycle. If k is exactly 1, the reactor is said to be critical if / < 1, it is subcritical if / > 1, it is supercritical. The neutron population and the reactor power depend on the difference between k and 1, ie, bk = k — K closely related quantity is the reactivity, p = bk jk. i the reactivity is negative, the number of neutrons declines with time if p = 0, the number remains constant if p is positive, there is a growth in population. 

The penetration theory predicts that should vary by the square root of the molecular difriisivity, as compared with film theoiy, which predicts a first-power dependency on D. Various investigators have reported experimental powers of D ranging from 0.5 to 0.75, and the Chilton-Colburn analogy suggests a 2/3 power. 

As shown by Table 14-12, empirical correlations for two-fluid atomization show dependence on high gas velocity to supply atomizing energy, usually to a power dependence close to that for turbulent breakup. In addition, the correlations show a dependence on the ratio of gas to liquid and system dimension. 

Apart from drese intrinsic properties, extrinsic effects can be produced in many oxides by variation of die metal/oxygen ratio drrough control of die atmospheric oxygen potential. The p-type contribution is increased as die oxygen pressure increases, and die n-type contribution as die oxygen pressure decreases. The pressure dependence of drese contributions can usually be described by a simple power dependence dins... 

Experiments show that emissive power depends on fireball size. Moorhouse and Pritchard (1982) present a graph of the relationship of fireball size and emissive power from results obtained by several investigators, among them, Hasegawa and Sato data from both 1977 and 1987. Figure 6.8 presents the Moorhouse and Pritchard (1982) graph to which the data from Johnson et al. (1990) have been added. 

When two atoms approach each other so closely that their electron clouds interpenetrate, strong repulsion occurs. Such repulsive van der Waals forces follow an inverse 12th-power dependence on r (1/r ), as shown in Figure 1.13. Between the repulsive and attractive domains lies a low point in the potential curve. This low point defines the distance known as the van der Waals contact distance, which is the interatomic distance that results if only van der Waals forces hold two atoms together. The limit of approach of two atoms is determined by the sum of their van der Waals radii (Table 1.4). 

A+B L -fl/2) have also been used. The theoretical assumption underlying an inverse power dependence is that the basis set is saturated in the radial part (e.g. the cc-pVTZ ba.sis is complete in the s-, p-, d- and f-function spaces). This is not the case for the correlation consistent basis sets, even for the cc-pV6Z basis the errors due to insuficient numbers of s- to i-functions is comparable to that from neglect of functions with angular moment higher than i-functions. 

The correlation energy is expected to have an inverse power dependence once the basis set reaches a sufficient (large) size. Extrapolating the correlation contribution for n = 3-5(6) with a function of the type A + B n + I) yields the cc-pVooZ values in Table 11.8. The extrapolated CCSD(T) energy is —76.376 a.u., yielding a valence correlation energy of —0.308 a.u. 

The product of force F and the rolling radius (R) of the tires on the drive wheels is the wheel torque (T). Power depends on both torque and rotational speed (N). By definition, power is given by P = 2tiNFR = 27tNT. Wlien driving at constant speed, the driver adjusts the accelerator pedal so the drive-wheel power exactly matches the power required (P,) to overcome the resistance of the vehicle (discussed later in this article). To accelerate the vehicle, the driver further depresses the accelerator pedal so that the power available at the drit c wheels (PJ exceeds P,.. 

In this investigation (Table VIII), it was found that Kw values for CO hydrogenation depend on the 0.9 power of the reciprocal of particle diameter. In view of this and the literature, a linear (first power) dependence on the reciprocal of particle diameter was used in the Kw expression. Accuracy of measurement is certainly insufficient to distinguish between a 0.9 and a 1.0 power dependence. 

Here, the chains are expected to be stretched, as indicated by the 2/3-power dependence of L on N, but less strongly than in solvent. The experimental evidence available to examine this argument is discussed in the section on block copolymer melts. 

The net equation contains no kinetic information. One cannot infer that the reaction rates (v) are given as vi = [Fe2l-]2[T13+] or v2 = [ArCl][R2NH]2. Although the rates will almost certainly depend upon the concentrations of the reactants, or at least on one of them, these particular power dependences are not required. The actual form must be determined experimentally. Unlike the situation in thermodynamics, the concentration exponents in the expression for the rate of reaction are not predictable from the net chemical reaction. 

The low TTA dependence at 35.0°C probably is attributable to dissolution of TTA in the aqueous phase. Observation of fourth-power dependence on acidity argues against any change in the extraction mechanism (e.g., Pu(IV) reduction or NO3 involvement). An aqueous Pu(TTA)3+ complex has been reported (14, 15) and this possibility has been considered in the error analysis of the Pu(IV)-sulfate stability constants. 

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