$\huge{H}\small{(start)} \huge{= 4.179 \frac{J/g}{C°} * mass(g) * 125C°}$
$\huge{H}\small{(generated)} \huge{= 9,806,160 \frac{J}{s}}$
$\huge{H}\small{(removed)} \huge{= \int_0^x\frac{4.179 \frac{J/g}{C°} * 2000g * (Temperature(C°) - 95C°) * \frac{9}{10}}{s} \,dx}$
$\huge{Temperature(C°) = \huge\frac{H\small{(start)} \huge{+} \huge H\small{(generated)} \huge{* x} - H\small{(removed)}}{4.179 \frac{J/g}{C°} * mass(g)}} $
What is the smallest value for $mass(g)$ that guarantees: $Temperature(C°) < 140$ on the interval $[0s, 53s] $
The system this is being applied to works as such: A continuous amount of Joules are being added to a mass with a starting temp of 125. That mass is also being cooled off and the rate at which it is cooled depends on the temp of the mass. This cycle lasts 53 seconds and if the temp of the mass exceeds 140 then the system will fail. These calculations assume an even temp throughout the mass.
I've checked all of these functions to the best of my abilities but it is possible I have formatted the integral incorrectly, if something does not seem to make logical sense please point it out and I will correct it.