Abstract: Reports in the literature of a “overshoot phenomena” during transient heat conduction in a finite slab in response to a step change in the boundary is studied systematically and why the overshoot is seen in some systems and not in others is explained thoroughly. A conclusion is drawn that the “overshoot pheonemena” is more a mathematical artifact rather than depiction of transient heat transfer events that can be expected in practice.
The Taitel paradox is revisted. The initial accumulation time condition was evaluated side by side for values of 0 initial accumulation, when initial accumulation is equal to the dimensionless group –Ve, Venrnotee number respectively for a finite slab subject to the constant wall temperature boundary condition. For a material with large relaxation time the overshoot was found for the model results when zero initial accumulation time condition. For the same set of material and parameters when a physically reasonable time accumulation condition was used the overshoot disappears. However, the transient temperature was found to be subcritical damped oscillatory. A steady state temperature was attained after a said time. This is different from the attainment of steady state only in the asymptotic limit of inifinite time in the Fourier model of heat conduction.
Lumped analysis was further explored. The average temperature in a finite slab subject to convective heating was obtained using:
(i) Fourier parabolic model. The model solution rises monotonically to a constant asymptotic value as given by equation (44);
(ii) Hyperbolic model with the first derivative of temperature with respect to time set to zero by the method of Laplace transforms. This model solution appears to have an overshoot as shown in Figure 5;
(iii) Hyperbolic model with the initial temperature at and the additional constraint that the average transient temperature should obey the energy balance equation from a lumped analysis.
The dimensionless temperature was expressed as a sum of a steady state temperature and a transient temperature. The transient temperature was expressed as a product of wave temperature and decaying exponential in time. The model solution was derived and illustrated in figures. As shown in Figure 6 the model solution does not exhibit any overshoot.
It appears that the damped wave conduction and relaxation equation can be applied to transient heat conduction problems without violation of the second law of thermodynamics. The storage number appears to be an important parameter in determination of the average transient temperature in a finite slab during damped wave conduction and relaxation. Expression for the time taken to attain steady state was derived. The maxima in the transient temperature were found to increase with decreasting starting with large values such as 10. A cross-over was found after became less than about 2.2. Then the maxima in the average transient temperature was found to decrease with decrease in storage number, Equation (66) can be seen to go to zero in the asymptotic limits of infinite storage number, where is the dimensionless storage number (Sharma Number). |