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This is not a homework question, this is a real question that I have.

I like to drink green tea, and I use an electric kettle which brings the water to boiling. You can purchase kettles which bring the water to a specific temperature, but I do not have one of those. I use a "normal sized" coffee mug which is made of the "normal material" (I am pretty sure it is ceramic). I am supposed to wait until the temperature of the water is 170 degrees Fahrenheit before I steep the tea. Assuming my house is at room temperature, how long should I wait?

In general I have the sense that temperature decays exponentially, but I don't know how to calculate the characteristic time $\tau.$ https://en.wikipedia.org/wiki/Newton%27s_law_of_cooling

$$T(t) = T_{env} + (T(0) -T_{env})e^{-t/\tau}$$ Here $T_{env}$ would be room temperature, $T(0)$ would be boiling, and I would want to set $T(t_{steep})=170$. On Wikipedia it says $\tau = C/hA,$ where $C$ is the heat capacitance, $h$ is the heat transfer coefficient, and $A$ is the surface area. I could estimate the surface area easily enough but am not sure about $C$ and $h.$ Should I just try to find the values of these for ceramic?

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    $\begingroup$ "I am supposed to wait until the temperature of the water is 170 degrees Fahrenheit before I steep the tea." - unrelated to the question, but according to who? Maybe green tea is different but black tea should be made with water as hot as possible. $\endgroup$
    – jacob1729
    Commented Dec 4, 2020 at 17:54
  • $\begingroup$ @jacob1729 That's what it says on the tea package itself. I think it depends on the type of green tea though, check this out thespruceeats.com/tea-brewing-temperature-guide-766367 $\endgroup$
    – Jbag1212
    Commented Dec 4, 2020 at 17:57
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    $\begingroup$ Sometimes it is easier to measure than to calculate. $\endgroup$
    – G. Smith
    Commented Dec 4, 2020 at 17:59
  • $\begingroup$ Okay, well the easiest way to get a controlled temperature of water will be to add cold water until it cools to the required temperature, buy an electric thermometer and a magnetic stirrer for more precision. $\endgroup$
    – jacob1729
    Commented Dec 4, 2020 at 17:59
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    $\begingroup$ There is a site on SE called Seasoned Advice where you might get more useful advice. They do deal with quite precise issue like this in the preparation of food. I agree with G Smith and others that measurement is the way to go here - theory won't be better than a rough estimate anyway. $\endgroup$ Commented Dec 4, 2020 at 18:20

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$$T(t) = T_{env} + (T(0) -T_{env})e^{-t/\tau}\tag{1}$$

To find $\tau$ we need to 'unpack' Newton's Law of Cooling a little. The law states:

$$\dot{Q}=hA[T(t)-T_{env}]$$

(check your link for the meaning of the symbols)

We also know that:

$$\dot{Q}=-mc_p\frac{\text{d}T(t)}{\text{d}t}$$

Combined we get:

$$\frac{\text{d}T(t)}{\text{d}t}=-\frac{hA}{mc_p}[T(t)-T_{env}]$$

Often we then state that:

$$\frac{1}{\tau}=\frac{hA}{mc_p}$$

Integration then yields $(1)$.

$A$, $m$ and $c_p$ are easy to determine but $h$ is harder. Usually tabulated values of $h$ for different situations can give a crude estimate.

$\tau$ can also be determined experimentally from relatively simple home experiments.


An easier and more accurate way of controlling temperature may be by mixing water of two known temperatures in the correct ratio.

For example, say we have a mass $m_H$ of boiling hot water (at $T_H$). To it we now add a mass $m_C$ of cold water (for example iced water at $T_C$).

Assuming the mixing was adiabatic, the final temperature $T_F$ is then approximately given by:

$$m_HT_H+m_CT_C\approx (m_H+m_C)T_F$$

This formula can be used to determine $m_C$ as a function of all other variables $T_F$, $m_H$, $m_C$ and $T_C$.

You could introduce an additional constraint, like:

$$m_H+m_C=M$$

where $M$ would be the full content of a vessel, like a mug or a teapot.

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  • $\begingroup$ I hadn't even thought of mixing cold water; that is a good idea, and it means I have to boil less water so it is faster. $\endgroup$
    – Jbag1212
    Commented Dec 4, 2020 at 18:57

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