Hyperbolic conduction

Figure 8 The comparison of temperature distribution in a semi-infinite solid based on Fourier s conduction and hyperbolic conduction approach...

Figure 8.5 shows the normalized temperature distribution inside the semi-infinite solid after imposition of the temperature pulse from both Fourier s conduction and hyperbolic conduction... 

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). 

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... 

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... 

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

In glycerol monooleate/decane bilayers we find the steady-state conductance at zero current to be proportional to the first power of the ion concentration and to the second power of the ionophore concentration, as illustrated in Fig. 1. (The current-voltage characteristic is hyperbolic for all ionic species indicating that this molecule is in the equilibrium domain for the interfacial reactions, with the rate-limiting step being the ion translocation across the membrane interior.) The conductance selectivity sequence is seen to be Na>K>Rb>Cs, Li. 

In practice, the values of coefficients A, B and C can be determined by conducting several experiments for the same solute at different flow rates. The method of multiple linear regression is then used to find the hyperbolic function that best matches the experimental values. 

Similarly, Figure 6b summarizes conductivity results. In contrast with pH, only conductivity measured in the first fractions (up to L/S 0.5 for BA and L/S 2 for 2SL) was of the same order of magnitude as that observed in the prerequisite study (Tab. 5). Moreover, conductivity measured in BA leachates, as well as in 2SL leachates, depicted a hyperbolic relationship with L/S ratio and showed marked... 

J. I. Frankel and B. Vick, General formulation and analysis of hyperbolic heat conduction in composite media, Int. J. Heat Mass Transf. 30, 1293-1305 (1987). 

A. Barletta and E. Zanchini, Hyperbolic heat conduction and thermal resonances in a cylindrical solid carrying a steady-periodic electric field, Int. J. Heat Mass Transf. 39, 1307-1315 (1996). 

A. Haji-Sheikh, W. J. Minkowycz, and E. M. Sparrow, Certain anomalies in the analysis of hyperbolic heat conduction, ASME J. Heat Transf. 124, 307-319 (2002). 

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. 

This paper deals with thermal wave behavior during frmisient heat conduction in a film (solid plate) subjected to a laser heat source with various time characteristics from botii side surfaces. Emphasis is placed on the effect of the time characteristics of the laser heat source (constant, pulsed and periodic) on tiiermal wave propagation. Analytical solutions are obtained by memis of a numerical technique based on MacCormack s predictor-corrector scheme to solve the non-Fourier, hyperbolic heat conduction equation. 

Heat waves have been theoretically studied in a very thin film subjected to a laser heat source and a sudden symmetric temperature change at two side walls. The non-Fourier, hyperbolic heat conduction equation is solved using a numerical technique based on MacCormak s predictor-corrector scheme. Results have been obtained for ftie propagation process, magnitude and shape of thermal waves and the range of film ftiickness Mid duration time wiftiin which heat propagates as wave. 

Baumeister, K.J. and Hamill, T.D. (1969) Hyperbolic heat conduction equation - a solution for the semiinfinite body problem. Journal of Heat Transfer, Vol. 91, pp. 543-548. 

Glass, D. E. Ozisik, M. N. and Vick. B. (1985) Hyperbolic Heat Conduction with Surface Radiation, International Journal ofHeatandMass Transfer, Vol. 28, pp. 1823-1830. 

Kar, A., Chan, C.L. and Mazumuder, J. (1992) Comparative Studies on Nonlinear Hyperbolic Heat Conduction for Various Boundary Conditions Analytical and Numerical Solutions, Journal of Heat Transfer, Vol. 114, pp. 14-20. 

Lewandowska, M. (2001) Hyperbolic Heat Conduction in the Semi-Infinite Body with a Time-Dependent Laser Heat Source,J. Heat Mass Transfer, Vol. 37, pp.333-342. 

Vick, B., and Ozisik, M.N. (1983) Growth and Decay of a Thermal Pulse Predicted by the Hyperbolic Heat Conduction Equation, Journal of Heat Transfer, Vol. 105, pp. 902-907. 

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