# Cup of hot tea, and I need it just *warm*

• Cantstandit
In summary, the conversation discusses the speaker's dislike for hot drinks and their curiosity about how long it would take for a cup of hot tea to cool down to a desired temperature. They mention an equation for heat flux and question if there is a non-iterative way to calculate it. Another person suggests using a differential equation to model the temperature over time.

#### Cantstandit

Hello,
Say I don't like to drink hot tea, in fact I don't like to drink anything hot. It is beyond my comprehension why anyone would want to drink something above 40C ;)

I have cup of hot tea 100C and i want to know how long will it take to cool down to 40C.

$$\frac{\Delta Q}{\Delta t}$$ = -k A $$\frac{\Delta T}{\Delta x}$$

(for some reason I can't get the equation to display correctly)
so i have k of course, "delta x" would be the thickness of the mug wall, A is also given . The "delta T" part is the difference between tea and air temperature, i guess?
Now my approach would be to write an iterative algorithm, that will calculate the "instantaneous" heat flux, then lower the temperature a bit and advance time, then calculate heat flux again. Unless there is some other equation which will let me calculate it in a non-iterative way?
It seems silly that I have to iterate just to get an answer to such simple question...
This is of course if I neglect the evaporation effect, say the mug is closed, it is a fancy mug.

In the limit of infinitesimal time steps time, your equation can probably be rewritten as a differential equation for the temperature like this:

$$\frac{dT}{dt}(t) = -b[T(t) - T_a]$$

where $$b$$ is some constant related to your $$a$$, $$k$$ and $$\Delta X$$, and $$T_a$$ is the ambient air temperature around the cup (assumed constant), you can integrate it to get

$$T(t) = [T(0) - T_a] \exp(-bt) + T_a$$

Which tells your the temperature at any time. $$T(0)$$ is the tea temperature at time 0.

## 1. What is the optimal temperature for a cup of hot tea?

The optimal temperature for a cup of hot tea is generally considered to be between 160-185°F (71-85°C). This allows the flavors and aromas of the tea to develop without scalding the tongue.

## 2. How long should I wait for my cup of tea to cool down to just warm?

The time it takes for a cup of tea to cool down to just warm can vary depending on the initial temperature of the tea and the surrounding temperature. However, on average, it takes about 5-10 minutes for a cup of tea to cool down to just warm.

## 3. Does the type of tea affect the ideal temperature for drinking?

Yes, different types of tea require different temperatures for optimal flavor. For example, delicate green teas are best enjoyed at lower temperatures (around 160°F or 71°C), while black teas can be enjoyed at slightly higher temperatures (around 185°F or 85°C).

## 4. Can I use a thermometer to check the temperature of my tea?

Yes, you can use a thermometer to check the temperature of your tea. However, it is important to remember to stir the tea before taking the temperature as the temperature can vary within the cup. Also, be sure to use a food-safe thermometer to avoid any contamination.

## 5. How can I cool down my tea quickly without diluting it with ice?

If you want to cool down your tea quickly without diluting it with ice, you can try placing the cup in a bowl of cold water or using a cooling stick designed specifically for tea. Another method is to pour the tea back and forth between two cups, which will help aerate and cool down the tea.