Simple Thermodynamic Model of Thermostats for a Liquid into a Tank: an Analytical Approach

Abstract

In this manuscript, we propose a simplified mathematical model based on the heat transfer laws to predict the temperature profiles of a liquid controlled by a simple thermostat. The model result in a set of linear ordinary differential equations ODEs with forcing which turn on and off at a priori unknown times TM = {ζ0, ζ1, . . . , ζM}. The p-th switch-time ζp ∈ Tp is calculated from the zeros of a function Q(χ) = Q(χ; ζ1, . . . , ζp−1) coming from analytical solutions of the system depending on the previous times ζ1, . . . , ζp−1. The mathematical problem can be solved by using standard techniques for solving ODEs once TM is calculated by M-successive iterations of the conditional expression Q(χ = ζp) = 0 and the Newton-Raphson method. We provide analytical expressions for the temperature as a function of time and TM considering direct (DC) and alternate (AC) feeding voltages. We solve the system using this numerical-analytical approach and compare it with the results of the 4th Runge–Kutta method finding a good agreement between both methods.