Water Cooling Calculator

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!

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!

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:

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.

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:

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!

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:

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

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

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:

The formula for $k$ is:

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

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!).

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!