# Water Cooling Calculator

Created by Rita Rain, Dominik Czernia, PhD and Davide Borchia
Reviewed by Steven Wooding
Last updated: Jan 30, 2023

Do you know you can calculate the cooling of the water in your cup of tea? Keep reading our article to learn:

• The physics of hot and cold beverages: a short introduction that will help you understand the problem.

• How to cool water to any temperature: calculate the cooling of water in three ways:

• Patience;

• Mixing cold and hot water; and

• Repeated transfers.

• An example calculation of how long it takes for boiling water to cool to a desired temperature.

And much more. Prepare your cup of tea, and let it cool while you learn how to calculate the cooling of water using science!

## The physics of a piping hot cup of tea

What is heat? Good question. As a concept we are in contact with every day, heat had multiple explanations from antiquity to modern theories. While in the past, scientists were way off (we are talking of fire everywhere and heat fluids!), modern physics has a fairly good explanation of heat.

Heat is the energy that "moves" when two bodies with different temperatures are in contact. The effect of heat is a change in temperature. Currently, our best understanding of this concept requires a look at the microscopic nature of matter: we describe heat as a transfer of kinetic energy of the molecules of a substance (either transitional, vibrational, or both). The higher the dissipation, the greater the heat our body is "releasing".

Close to the absolute zero ($\small -273.15\ \degree\mathrm{C}$), the kinetic energy of particles reduces to zero. All particles collapse to the ground state (the lowest energy state they can live in), but the rules of quantum mechanics prevent all of them from doing so! This is why we can't reach exactly that temperature. In a Bose-Einstein condensate, a huge fraction of particles reach the ground state: this way, scientists, managed to reach temperatures a fraction of a millionth of degrees above absolute zero!

To understand how fast water cools and the possible ways to do so, we can ignore this detailed description of heat. However, knowing that our target is to dissipate kinetic energy, maximizing the heat transfer in the process, it will be helpful to grasp the physics of the next section!

## Cooling water to any temperature: calculate how you long for water to cool down

The physics of the last section won't be necessary to understand the formulas we present in this part of the article. As in many other cases, we can describe thermodynamics using macroscopic empirical formulas. So, worry not, and prepare to explore the calculations for water cooling!

#### Cool down water by waiting

How does kinetic energy transfer? Through collisions, both between particles and between particles and the side of the container. Each one of the collisions causes a bit of the total energy of the system to be transferred and eventually exit the system when the collision happens in the "right direction".

Luckily for us, we don't need to analyze the motion of every single particle to understand how the kinetic energy of your tea moves from inside the cup to outside it. Isaac Newton, well before modern physics explained what heat is, described this process with his cooling law, which only requires you to know the macroscopic characteristics of your cup! Here is the formula:

$\small t = \cfrac{-\ln\left[\cfrac{(T - T_\text{ambient})}{(T_\text{initial} - T_\text{ambient})}\right]}{k}$

where:

• $T$ — The desired final temperature;
• $T_\mathrm{initial}$ — The starting temperature. For boiling water at sea level, take $T_\mathrm{initial} = 100\ \degree\mathrm{C}$;
• $T_\mathrm{ambient}$ — The temperature of the room where you're preparing your tea; and
• $k = h \cdot A / C$ – The cooling coefficient. We calculate it from the heat transfer coefficient, $h = 284.16 \text{ W/(m}^2\! \cdot\! \text{K})$ (obtained from experiments), the area of heat exchange, $A$, and the heat capacity, $C$ (if you don't remember how to find this value, head straight to our heat capacity calculator).

The result of this formula is the time required to reach the desired final temperature, $t$.

Visit Omni's Newton's law of cooling calculator for more insights on the cooling process!

As you can see, how fast water cools depend on the quantities above in different ways:

• The higher the initial temperature, or the lower the final temperature, the longer the time required to cool the water.

• The higher the temperature of the environment, the longer the time.

• The bigger the surface where the heat exchange happens, the shorter the time: A low, flat cup will cool faster than an almost spherical one.

🙋 When the difference between the initial and ambient temperature is too high, the observed results deviate from the ones expected by the formula detailed above, and the cooling proceeds faster. This calculator implements only the theoretical formula: to find the corrections, try to experiment at home!

#### Cooling water by adding cold water to the cup

You can reduce the average kinetic energy of the particles in your cup by adding colder water. This way, even if it remains inside the cup, the desired transfer of heat still happens: hotter, faster molecules collide with slower ones, transferring their heat. The formula to describe this process is surprisingly easy:

$\small V_\text{cold} = \frac{V_\text{total} \, (T_\text{initial} - T)}{(T_\text{initial} - T_\text{cold})}$

Part of this formula can remind you of the previous one: a ratio of two temperature differences. The result, $V_\mathrm{cold}$, is the approximate volume that can bring the total amount of water (cold plus hot) to the desired temperature $T$.

#### Cooling water by transfer

The last method you can use to cool down your tea is to repeated transfers between two glasses. Doing so, you increase the surface exposed to the air, allow the glasses to cool partially, and favor more collisions between particles. The net effect is a fast dissipation of heat, but this plethora of mechanism make a description, even empirical, of the process pretty hard: trust us on this, and use the third option of our water cooling calculator, and for once, forget the math!

## Calculate how long does boiling water take to cool: an example

Take a cup of nearly boiling water; it can be green tea, for example. Under standard conditions, the liquid has a temperature of $90\ \degree\mathrm{C}$ ($194\ \degree\mathrm{F}$). How long does it take to the water to almost reach room temperature, let's say, $20\ \degree\mathrm{C}$ ($68\ \degree\mathrm{F}$)?

We can use Newton's cooling law:

$\small t = \cfrac{-\ln\left[\cfrac{(T - T_\text{ambient})}{(T_\text{initial} - T_\text{ambient})}\right]}{k}$

Where we perform the following substitutions:

• $T_\mathrm{initial} = 90\ \degree\mathrm{C} = 363.15\ \mathrm{K}$;

• $T= 21\ \degree\mathrm{C} = 294.15\ \mathrm{K}$; and

• $T_\mathrm{ambient} = 20\ \degree\mathrm{C} = 293.15\ \mathrm{K}$;

🙋 We needed to make the final temperature a degree higher than the ambient temperature to avoid reducing the argument of the logarithm to $0$: mathematically speaking, to reach exactly room temperature, the cups need an infinite amount of time!

The value of $k$ depends on the heat transfer coefficient, $k$ the area of heat exchange, $A$, and the heat capacity, $C$:

• $k = 284.16 \text{ W/(m}^2 \cdot \text{K})$;

• $A = 0.005027\ \mathrm{m^2}$ for an average cup; and

• $C = c\cdot \rho \cdot V$, where we substitute the values for specific heat and density of water, and consider a $400\ \mathrm{ml}$ cup: $C = 4200\ \mathrm{J/(K\cdot kg)}\cdot 973\ \mathrm{kg/m^3}\cdot 0.0004\ \mathrm{m^3}=1634.64\ \mathrm{J/K}$

The formula for $k$ is:

$\small \begin{split} C &= 4200\ \mathrm{J/(K\!\cdot\! kg)}\!\times\! 973\ \mathrm{kg/m^3}\\ &\qquad \times0.0004\ \mathrm{m^3}\\&=1634.64\ \mathrm{J/K} \end{split}$

The formula for $k$ is:

$\small\begin{split} k &= h \cdot\frac{A}{C}\\ &\!\!= 99.55 \text{ W/(m}^2\! \cdot\! \text{K})\frac{0.005027\ \mathrm{m^2}}{1634.64\ \mathrm{J/K}}\\ &\!\!=0.0008738\ \mathrm{1/s} \end{split}$

Now you have all the values required: substitute them in the cooling formula:

$\small \begin{split} t \!&=\! \cfrac{-\ln\left[\cfrac{(21\ \mathrm{\degree C} \!-\! 20\ \mathrm{\degree C})}{(90\ \mathrm{\degree C} - 20\ \mathrm{\degree C})}\right]}{0.0008738\ \mathrm{1/s}}\\[1em] \!&\approx\!4862\ \mathrm{s} \end{split}$

It would take slightly more than $80$ minutes for your tea to reach room temperature: but don't wait for so long (unless you like iced tea!).

## How to use our water cooling calculator

Choose the desired type of cooling, and insert the desired values: be careful in choosing the proper temperatures in the correct fields and the appropriate volumes. We will give you the time, volume, or the number of transfers needed to reach the desired temperature. However, remember that real-world physics is slightly different from blackboard physics: the behavior you'll observe may differ somewhat from what you calculated.

And if you want to know more than just physics, head to our tea brewing calculator for tips and tricks to obtain the perfect tea!

## FAQ

### What is the Mpemba effect?

The Mpemba effect is an observed physical phenomenon in which hot (even boiling) water freezes faster than cold water (under the same conditions). While the existence of the phenomenon is proven, scientists still argue about its mechanism, with explanations ranging from the enhanced movement of ice crystals to reduced energy requirement for the process (fewer bonds present in boiling water, higher concentration of ice-like precursors).

You can try this experiment at home, but be careful about putting boiling water in a freezer!

### How do I calculate the cooling time of water?

To calculate the cooling time of water to a desired temperature, use the following formula from Newton's cooling law:

t = -ln[(T final - T ambient)/(T initial - T ambient]/k

where:

• T initial and T final — The starting and desired temperatures of water;
• T ambient — The temperature of the room; and
• k — A constant that contains information on the shape of the container and the physical characteristic of water. For a 300 ml cup with a diameter of 6 cm, k = 0.000655.

### How do I cool boiling water quickly?

To cool boiling water quickly, you can use one of three methods:

• The longest one: waiting some time: hot water cools pretty fast, so if you want warm water, this may be your best option!

• The most accurate one: adding cold water. Adding a specific amount of cold water at a known temperature allows you to reach the desired temperature with little error.

• The fastest one: transfer multiple times the water from one glass to another: you will be surprised by how fast it cools down!

### How long does a cup of 90 °C water take to reach 75 °C?

Assuming a 300 ml cup with a diameter of 8 cm in a room at 24.6 °C, the time would be around 3 minutes and 43 seconds. To calculate the result:

1. Calculate the difference between the desired and the room temperature: 75 - 24.6 = 50.4 °C.

2. Calculate the difference between initial and room temperature: 90 - 24.6 = 65.4 °C.

3. Calculate the negative natural logarithm of the ratio of the previous results: -ln(50.4/65.4) = 0.26.

4. Divide the result by the constant k = 0.001165.

The result is 223 seconds, or just below 3 minutes, 43 seconds.

Rita Rain, Dominik Czernia, PhD and Davide Borchia
Method
Wait some time
Ambient temperature
°F
Initial temperature
°F
Desired temperature
°F
Volume
cups
Cup diameter
in
Waiting time
1
min
1
sec
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