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 $${\mathcal {T}}_M=\left\{ \zeta _0,\zeta _1,\ldots ,\zeta _M\right\} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>M</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mfenced> <mml:msub> <mml:mi>ζ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ζ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ζ</mml:mi> <mml:mi>M</mml:mi> </mml:msub> </mml:mfenced> </mml:mrow> </mml:math> . The p th switch-time $$\zeta _p\in {\mathcal {T}}_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ζ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:math> is calculated from the zeros of a function $${\mathcal {Q}}(\chi )={\mathcal {Q}}(\chi ;\zeta _1,\ldots ,\zeta _{p-1})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>χ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>χ</mml:mi> <mml:mo>;</mml:mo> <mml:msub> <mml:mi>ζ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ζ</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> coming from analytical solutions of the system depending on the previous times $$\zeta _1,\ldots ,\zeta _{p-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ζ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ζ</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> . The mathematical problem can be solved by using standard techniques for solving ODEs once $${\mathcal {T}}_M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>M</mml:mi> </mml:msub> </mml:math> is calculated by M -successive iterations of the conditional expression $${\mathcal {Q}}(\chi =\zeta _p)=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>(</mml:mo> <mml:mi>χ</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ζ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and the Newton–Raphson method. We provide analytical expressions for the temperature as a function of time and $${\mathcal {T}}_M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>M</mml:mi> </mml:msub> </mml:math> 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.