Entropy Production Under Unsteady-State Thermal Conditions with a Temperature Gradient

In this study, the behavior of entropy production per unit volume for an unsteady-state thermal regime with a temperature gradient in the bodies of a simple shape (unbounded plate, sphere, and unbounded cylinder) is investigated to further understand the aspects of the linear regime of thermodynamics under complex heat exchange conditions. The change in the production of entropy per unit volume of the simple-shaped bodies under the specified conditions is investigated. The known analytical solutions to the one-dimensional problems of heating these bodies under boundary conditions of type 2, obtained in the approximation of constant properties, are used. The total value of entropy production per unit volume is calculated as the sum of the time-constant steady-state component owing to the temperature difference and the unsteadystate component determined in the absence of a temperature gradient. The contribution of the temperature drop across the plate thickness to the total value of entropy production per unit volume is estimated through the product of the force and the corresponding heat flux, and this estimate is extended to the cases of a sphere and cylinder towing to the equality of the temperature drop in all three bodies. The unsteady-state component of entropy production per unit volume is calculated using the logarithm of the ratio of two instantaneous temperatures divided by the difference between the corresponding time values. The unsteadystate component of entropy production per unit volume is demonstrated to correspond to the extremum principle as the Fourier number increases. A comparison of the unsteady-state components of entropy production per unit volume of the plate, sphere, and cylinder shows that the extremum principle is more pronounced for the sphere. The results enable to expand the understanding of the theory of the linear mode of thermodynamics.