Background: I have a problem where a temperature probe is submerged into a fluid at a higher unknown temperature. Ambient temperature is also unknown. The temperature at three different times is known, and I am assuming a 1st order response system.
Problem: I have 3 equations and 3 Unknowns, but this is a non-linear system. How do I solve for the 3 unknowns ($\tau, T_\infty, T_0$)?
$T_\infty + (T_0 - T_\infty)e^{\frac{-1}{\tau}}=56$
$T_\infty + (T_0 - T_\infty)e^{\frac{-2}{\tau}}=136$
$T_\infty + (T_0 - T_\infty)e^{\frac{-5}{\tau}}=209$
Thanks in advance
Let $z = e^{-1/\tau}$, $x = T_0 - T_\infty$, so the equations become
$$ \eqalign{T_\infty + x z &= 56\cr T_\infty + x z^2 &= 136\cr T_\infty + x z^5 &= 209\cr} $$ The first equation says $T_\infty = 56 - x z$. Substitute that into the other two:
$$ \eqalign{x (z^2 - z) &= 80\cr x (z^5 - z) &= 153\cr} $$ Divide the second equation by the first:
$$ \dfrac{x(z^5-z)}{x(z^2-z)} = z^3 + z^2 + z + 1 = \dfrac{153}{80} $$
Now solve this cubic equation. The real solution is $$ \eqalign{z &= \dfrac{1}{60} \left(126550+150 \sqrt{734529}\right)^{1/3} - \dfrac{40}{3 \left(126550+150 \sqrt{734529}\right)^{1/3}} - \frac{1}{3}\cr &\approx 0.5134705159}$$