Simple thermodynamic model of thermostats for a liquid into a tank: an analytical approach

• Published in: Journal of engineering mathematics Vol. 150; no. 1
• Main Authors: Salazar, Robert, Deaza, Felipe, Zamudio, José, García, Leonardo
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Nature B.V 01.02.2025
• Subjects: Differential equations; Exact solutions; Mathematics; Newton-Raphson method; Ordinary differential equations; Runge-Kutta method; Temperature profiles; Thermodynamic models; Thermostats
• ISSN: 0022-0833; 1573-2703
• DOI: 10.1007/s10665-024-10416-5

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 $${\mathcal {T}}_M=\left\{ \zeta _0,\zeta _1,\ldots ,\zeta _M\right\} $$ T M = ζ 0 , ζ 1 , … , ζ M . The p th switch-time $$\zeta _p\in {\mathcal {T}}_p$$ ζ p ∈ T p is calculated from the zeros of a function $${\mathcal {Q}}(\chi )={\mathcal {Q}}(\chi ;\zeta _1,\ldots ,\zeta _{p-1})$$ Q ( χ ) = Q ( χ ; ζ 1 , … , ζ p - 1 ) coming from analytical solutions of the system depending on the previous times $$\zeta _1,\ldots ,\zeta _{p-1}$$ ζ 1 , … , ζ p - 1 . The mathematical problem can be solved by using standard techniques for solving ODEs once $${\mathcal {T}}_M$$ T M is calculated by M -successive iterations of the conditional expression $${\mathcal {Q}}(\chi =\zeta _p)=0$$ Q ( χ = ζ p ) = 0 and the Newton–Raphson method. We provide analytical expressions for the temperature as a function of time and $${\mathcal {T}}_M$$ T M 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.