Relaxation time balance equations

Before we come to further determinations of the unknown quantities, we shall estimate here the effect of the internal angular momentum on the motion of the liquid. Let a be the characteristic size of internal structural elements, then Sik pav, <7ik pv/a, where p is the viscosity coefficient. An estimate of the characteristic relaxation time of the balance of the internal and external rotation follows from equation (8.7)... 

For times which are much bigger than the relaxation time, the internal and external rotation are balanced, so equation (8.7) is followed by... 

Models of the isothermal mechanism can be constructed using a balance equation (1) for the area of active surface per unit volume of a solid sample, with a term added which describes the propagation of this surface into the nonfractured matrix. The term requires that a certain effective transfer coefficient (analogous to the diffusion coefficient) should be introduced. To a first approximation, it can be written as D = vl = v2r, where v is the velocity of sound in the sample, / is the length of the free run of a crack for the time r, and t is the time of mechanical unloading (or the characteristic relaxation time of stresses in the real solid matrix of a reactant sample). It seems impossible to... 

With the exception of this relaxation method, all the methods described solve the stage equations for the steady-state design conditions. In an operating column, other conditions will exist at startup, and the column will approach the design steady-state conditions after a period of time. The stage material balance equations can be written in a finite difference form, and procedures for the solution of these equations will model the unsteady-state behavior of the column. 

In the general case the concentrations of the dimers and trimers cannot be neclected. This means that the number of the aggregates does not remain constant during the fast process. Therefore eq. (10) is not valid. The only relation between and g is the mass balance equation. In order to get a single relaxation time, a must be zero in eq. (25) and eq. (27)... 

Most chemical processes including micellar kinetics involve several steps and are characterized by several relaxation times (relaxation spectrum). The maximum number of observable relaxation times is equal to the number of independent rate equations that can be written for the system investigated. This number is equal to that of chemical species minus the number of mass-balance equations. > ... 

The end-to-end vectors of the subchains have distributions in their length and orientation. Equation [10] clearly indicates that the deviatoric part (measurable part) of the stress tensor due to the entropy elasticity of the polymer chains, hereafter referred to as the polymeric stress, reflects the orientational anisotropy of the subchains specified by the configuration tensor S(n,t). Consequently, the polymeric stress relaxes, even though the material keeps its distorted (e.g., sheared) shape, when the orientational anisotropy induced by tbe applied strain relaxes tbrougb tbe tbermal motion of tbe cbains. (In tbis relaxed state, S(n,t) is equal to 1/3 and tbe subcbain tension is transmitted isoUopically in aU tUrecrions to balance tbe isotropic pressure.) Thus, tbe relaxation time of the polymeric stress is identical to the orientational relaxation time of the polymer chains. 

Equation [78b] demonstrates that the intetfadal relaxation time tint is essentially determined by a balance of the driving force for the shape recovery of the droplet, r/J v/ and the viscous resistance, J/o.mT ... 

The most important characteristic of an ideal batch reactor is that the contents are perfectly mixed. Corresponding to this assumption, the component balances are ordinary differential equations. The reactor operates at constant mass between filling and discharge steps that are assumed to be fast compared with reaction half-lives and the batch reaction times. Chapter 1 made the further assumption of constant mass density, so that the working volume of the reactor was constant, but Chapter 2 relaxes this assumption. 

Here, yb.b is the rate of VR from state b to state b. yb is the total decay rate of level by, consisting of a VR part and a radiative part (yr). Equation (119c) is the detailed balance condition, which guarantees that the system will evolve to thermal equilibrium at long times. Finally, Eq. (119d) accounts for the decay of coherences due to the relaxation matrix. The relaxation matrix [Eqs. (119)] corresponds to the following equations of motion ... 

The Eqs. (5.210) and (5.211) describe the diffusion of monomers and micelles as before. The physical meaning of Eq. (5.212) is not so obvious. It can be regarded as the equation of the local balance of surfactants in micelles. If the initial distribution of micelles is homogeneous and only the monomer concentration is perturbed, the first relaxation proeess can lead to the dependence of the aggregation numbers on space coordinates and time even in absence of a concentration gradient of the total number of micelles. Surfactants can be transferred not only as a result of the monomer diffusion but due to the diffusion of aggregates of different aggregation numbers. This effect is described by equation (5.212). 

The word creep means literally a slow movement. In everyday Hfe this is usually from one point to another. However, in polymer technology the word is appHed in two rather different situations. If a high modulus polymer is instantaneously extended to a given strain, the stress that is engendered slowly reduces. This relaxation of the stress is called creep and the elements of stress relaxation are discussed further in Chapter 10. If a stress is apphed to a low modulus rubber, the material quickly extends to a length where the internal restoring force balances the external stress as described by the Mooney—Rivhn equation. However, if the stress is held constant, over a longer time the rubber may slowly extend further. 

Doi and Onuki [50] (DO), extended the models to polymer blends in which both components are entangled. The key aspect to address is how to incorporate stress into the equation of motion for concentration fluctuations. Effectively, by determining the conditions for force balance, it was shown that the stress enters the equation of motion at the same level as the chemical potential. Such an approach enabled the development of a framework that coupled the dynamics of concentration fluctuations to the flow fields and stress gradients however, only the simplest form of constitutive relation for the stress was treated. In entangled polymer solutions, the tube model predicts that the relaxation of an imposed stress is well described by a single exponential decay, with the characteristic time-scale being that required for... 

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