Damped Wave Diffusion and Relaxation

The semi-infinite medium is employed to study the spatiotemporal patterns that the solution of the non-Fick damped wave diffusion and relaxation equation exhibits. This medium has been used in the study of Pick mass diffusion. The boundary conditions can be different kinds, such as constant wall concentration, constant wall flux (CWF), pulse injection, and convective, impervious, and exponential decay. The similarity or Boltzmann transformation worked out well in the case of the parabolic PDF, where an error function solution can be obtained in the transformed variable. The conditions at infinite width and zero time are the same. The conditions at zero distance from the surface and at infinite time are the same. 

Baumeister and Hamill [32] solved the hyperbolic heat conduction equation in a semi-infinite medium subjected to a step change in temperature at one of its ends using the method of Laplace transform. The space-integrated expression for the temperature in the Laplace domain had the inversion readily available within the tables. This expression was differentiated using Leibniz s rule, and the resulting temperature distribution was given for x X as 

The method of relativistic transformation of coordinates is evaluated to obtain the exact solution for the transient temperature. Consider a semi-infinite slab at initial concentration Co, imposed by a constant wall concentration Q for times greater than zero at one of the ends. The transient concentration as a function of time and space in one dimension is obtained, yielding the dimensionless variables 

The mass balance on a thin spherical shell at x with thickness Ax is written in one dimension as -dJ ldX = du/dx. The governing equation can be obtained in terms of 

It can be seen that the governing equation for the dimensionless mass flux is identical in form with that of the dimensionless concentration. The initial condition is 

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Importance of mutual diffusion Phase separation, polymer blend processing Relation to interaction parameter Transient diffusion Pick s law of diffusion Damped wave diffusion and relaxation Semi-infinite medium, finite slab Periodic boundary condition... 

The mass balance on a thin spherical shell at x with thickness Ax is written. The governing equation can be obtained from eliminating J" between the mass balance equation as given in Equation (9.107) and the hyperbolic damped wave diffusion and relaxation equation as given in Equation (9.123). The governing equation can be rendered dimensionless and seen to be... 

Polymer Thermodynamics Damped Wave Diffusion and Relaxation... 

FIGURE 9.3 Concentration profile under damped wave diffusion and relaxation in semiinfinite medium. 

Transient diffnsion in a semi-infinite medium was studied under a constant wall concentration bonndary condition using Pick s second law of diffusion and the damped wave diffnsion and relaxation equation. The latter can acconnt for finite speed of propagation of mass. A new procedure called the method of relativistic transformation was developed to obtain bounded and physically realistic solntions. These solntions were compared with the solution from Pick s second law of diffusion obtained nsing the Boltzmann transformation and the solution presented in the literatnre by Baumesiter and Hamill [32]. Four different regimes of the solution... 

As shown in Fig. 19 for solid samples, monochromatic light, chopped at a frequency in the order of magnitude of 10-1000 cps which is low compared with the velocity of deactivation, strikes the solid sample contained in a sample holder. After excitation and relaxation the released heat diffuses to the surface, passes into the gas phase and acts as an acoustic piston which generates a pressure wave detected by the microphone and amplified by a phase-sensitive amplifier locked to the chopping frequency co. Solution of the heat diffusion equation proves that after a distance x from their starting point the heat waves are damped by ... 

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