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I'm having trouble with the following homework question!

A container is full of liquid at 185C and is cooled by another fluid which was maintained at a temperature of 15C. The temperature of the liquid fell by 75C in 20 minutes. By using Newton's rate equation determine how long ti takes for the liquid in the container to falling temperature to 50C.

Provide an approximation for the time it takes the liquid to stabilize in temperature to 15C.

I tried to calculate this myself and determined the constant to be $k = 0.0290961$. I got this by using:

$50-15=(110-15)e^{-20k}$

I then used this constant to determine how long it would take for it to fall to 50 C and I got 34.3183 minutes. I then tried to use the Fourier version which is I think used for approximations:

$\frac{dT}{dt} = -k(T-T_f).$

This got me a number very close to 34 minutes again...

The question in the second part asks to provide an approximation for the time taken, but I realise now I can't do this because this equation would just tell you infinity (as that really is how long it would take for it to fall to that temperature). My friend used the Fourier equation in both cases and his answers make more sense to look at (45 minutes for the approximate time to 15 C), but in my notes the one I used is labelled as the rate equation.

What am I doing wrong? Here is my working in full:

Part 1

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    The time constant is defined as $\tau=-1/k\approx 34.369$ min. In some applications a practical rule is to consider 5 time constants $5\tau \approx 172$ min, which corresponds to a final temperature of $T=15+(185-15)e^{-5}\approx 16.1$ C instead of 15C.2012-12-15

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If you have the expression for the temperature $T$ at time $t$, then for your first question just solve $T(t)=50$ for $t$.

For the second question, well, $T(t)\to 15$ as $t\to\infty$, so I'm not sure what they want other than that observation.

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    No offense was intended, but your answer is not clear enough for me at my skill level.2012-12-15