(79f) On the Sensitivity of "Overshoot" to Initial and Boundary Conditions and Engineering and Mathematical Perspectives

During scale up of chemical processes and control design sometime underdamped second order systems are encoutered. These systems can be expected to have a overshoot in response to a step change in the input.  One example is taken in this study is the non-Fourier damped wave conduction in a finite slab. The average temperature of the slab can be described using a second order system.  For certain values of the damping coefficient the overshoot in temperature can be a violation of second law of thermodynamics.    A careful side by side study of this problem is presented below.  The time condition  was assumed to be 0 in Taitel [1972].  Consider a finite slab of width 2a, at a initial temperature of T₀ heated by hot isothermal walls brought a temperature, T_s < T₀ for times t > 0, greater than 0.  As can be seen from Figure 3.1 below only the left hand limit of the rate of change of average temperature of the slab wth respect to time is zero.   A lumped analysis on the heating process leads to the expression for the rate of change of average temperature of the interior of the slab as shown in the ordinate of Figure 1.0. The right hand limit, of the rate of change of average temperature in the finite slab is a maximum!  With increase in time this decreases exponentially to a asymptotic zero at large times or upon attainment of steady state. The right hand limit was used in the analysis.  For a material with large relaxation time the overshoot was found for the model results when initial accumulation time condition is taken as 0. For the same set of material and parameters when a physically reasonable time accumulation condition was used the overshoot disappears. The transient temperature was subcritical damped oscillatory. A steady state temperature was attained after a said time. 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. (ii) Hyperbolic model with the first derivative of temperature with respect to time set to zero as used by Taitel [1972] by the method of Laplace transforms. This model solution appears to have an overshoot ; (iii) Hyperbolic model with the initial temperature at T₀ 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 is presented. S* appears to be an important parameter in the analysis. This is called "Sharma Number (heat)". S *is the dimensionless storage number.  As shown in Figure the model solutiond oes 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 Sharma 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 developed. The maxima in the transient temperature were found to increase with decreasing S* starting with large values such as 10.   A cross-over was found after S* became less than about 2.2.  Then the maxima in the average transient temperature was found to decrease with decrease in Sharma number, S*.   The average transient temperature becomes zero in the infinite limit of S*.