Hyperbolic Relaxation

Equation (2-3.7) suggests that at very low values of conductivity (/c 0.01 pS/m), charge will relax extremely slowly from a liquid. Eilters for example would have to be an hour or more upstream of tanks before the charge would dissipate to 5% of its initial value. 

TABLE (a). Experimental Data for Charge Density Q Downstream of a Microfilter 

TABLE (b). Charge Dissipation Times for Hyperbolic Relaxation (e, = 2-4) 

Owing to the change from exponential to hyperbolic relaxation, a nominal dissipation time of 100 s is assigned in Appendix B to all liquids whose conductivities are below 2 pS/m. The 2 pS/m demarcation is convenient 

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The hyperbolic relaxation equation (A-5-2.4.1 a) contains charge carrier mobility as a variable, which should be sensitive to oil viscosity. This is found to be the case for some viscous nonconductive liquids. These have much slower rates of charge dissipation equivalent to an Ohmic liquid whose conductivity is 0.02 pS/m (5-2.5.4). 

There are no available data to establish whether nonconductive, low viscosity chemical products such as ethyl ether similarly display hyperbolic relaxation below about 2 pS/m, or even whether this phenomenon is a practical reality for such liquids. Should Ohmic relaxation behavior continue to much less than 0.5 pS/m the risk of static accumulation would be enhanced compared with petroleum distillates. 

Table (a) shows experimental data [24] for the initial charge density exiting a fuel filter Qq plus the charge density Q remaining 30 s downstream. At low conductivity the charge decays much faster than predicted by an exponential relaxation law [Eq. (2-3.7)] and instead follows a hyperbolic relaxation law [24] given by... 

Assuming the validity of Adam-Gibbs equation for relaxation dynamics and the hyperbolic temperature dependence of heat capacity, the strength parameter is found to be inversely proportional to the change in heat capacity [see Eq. (2.10)] at the glass transition temperature [48,105]. 

Other research groups derived viscoelastic properties from creep experiments of the final tablet [150-154], As Tsardaka and Rees [142] determined, stress relaxation follows a hyperbolic equation. 

The above phenomena me physically miomalous and can be remedied through the introduction of a hyperbolic equation based on a relaxation model for heat conduction, which accounts for a finite thermal propagation speed. Recently, considerable interest has been generated toward the hyperbolic heat conduction (HHC) equation and its potential applications in engineering and technology. A comprehensive survey of the relevant literature is available in reference [6]. Some researchers dealt with wave characteristics and finite propagation speed in transient heat transfer conduction [3], [7], [8], [9] and [10]. Several analytical and numerical solutions of the HHC equation have been presented in the literature. 

If a system is uniformly hyperbolic, every point in phase space has both stable and unstable directions, and the maximum Lyapunov exponent with respect the maximum entropy measure is positive. The system has the mixing property and is therefore ergodic. The correlation function of observables also shows exponential decay. Uniformly hyperbolicity, which is sometimes rephrased as strong chaos in physical literature, is a well-established class of systems and is controllable by means of many mathematical tools [15]. In hyperbolic systems, there are no sources to make the relaxation process slow. 

In contrast to hyperbolic systems, the phase space structure in the mixed system is quite intricate and inhomogeneous, which brings about transport phenomena and relaxation processes essentially different from uniformly hyperbolic cases [3]. A remarkable fact is that qualitatively different classes of motions such as quasi-periodic motions on invariant tori and stochastic motions in chaotic seas coexist in a single phase space. The ordered motions associated with invariant tori are embedded in disordered motions in a self-similar way. The geometry of phase space then reflects the dynamics. 

Neglecting relaxation, the envelope of a strongly radiation-damped free induction decay (FID) is given by a hyperbolic secant function, ... 

This type of damping agrees with the hyperbolic-like relaxation function of DL, but also provides the frequency stability of every oscillation for time-independent... 

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