Serrin’s accumulation function, the First and the Second Laws of Thermodynamics

Abstract

Beginning in 1979, the First and Second Laws of Classical Thermodynamics were given a new lease of life through the researches of Serrin. The definition of a positive, empirical temperature scale was shown to depend on the ideas of Maxwell and Mach. Next, a thorough description of thermodynamical systems and product systems was given, and the internal energy was defined as a function which acts as a lower potential for the difference between the heat transferred and the work done for each process. Using simple mathematics and the First Law for cycles, the existence of the internal energy function for reversible systems was derived through a well known theorem from multivariable calculus. Further, the accumulation function which defines the heat transferred between the working substance and its surroundings up to and equal to a given temperature was introduced. From this, it emerged that Clausius’ inequality for a cycle was equivalent to an integral inequality involving the accumulation function over the open, positive real numbers. It was demonstrated that this inequality leads to the definition of the absolute temperature scale. From this, entropy was defined as an upper potential to the integral appearing in the Accumulation Theorem, for each process. The existence of the entropy function for reversible processes was also proved. In addition, it was shown by Huilgol that the First Law for a cycle can be represented by an inequality, which places bounds on the work done in a cycle. Moreover, the First Law and the entropy function derived from the Second Law can be used to derive an upper bound on the work done in a cycle which is sharper than that due to Carnot. Specifically, if the cycle is irreversible, the bound includes the product of the change in entropy over the adiabatic parts of the cycle, with the least value of the absolute temperature over the heat rejecting part. In the work presented here, these ideas will be explored with a view to incorporating them, along with newer versions of the Second Law due to Serrin, in the standard curriculum of thermodynamics. In order to facilitate this inclusion, explicit forms for the accumulation function in the Otto, Diesel, Stirling and Ericsson cycles are derived.