As fast as possible! 'Convective Heat Transfer Coefficient' to accelerate milk cooling

When making baby formula, you dissolve the powdered milk in hot water inside the baby bottle. Next, you cool the piping hot bottle by running it under tap water. In the middle of the night, you'll be thinking, 'I want this to cool down as fast as possible...'

At this time, if you spin the baby bottle around with your hand, for some reason it cools down faster than if you just leave it alone. If you have experience with childcare, many of you have probably felt this.

So, why does spinning it make it cool down faster?

It's not just because it mixes; actually, a parameter called convective heat transfer coefficient is involved. The physics we learned in school shows up in little life hacks... this time, that's the theme of this familiar science.

✔︎ Leaving it alone vs. spinning it results in different cooling

Even when exposed to running water in the same way, there is a big difference in how it cools down between leaving the bottle alone and spinning it. If you leave it alone, it cools down gradually from the outside of the bottle. Therefore, while the parts near the outside become lukewarm, the center tends to remain hot. When you think it's cooled down and try to feed the baby, you realize it's 'still hot...' and end up having to cool it again.

On the other hand, when you spin the baby bottle, the entire milk inside moves, mixing the hot parts with the cold parts. As a result, it starts to cool down all at once, not just on the outside but also in the center, eliminating temperature unevenness.

What is important here is that it doesn't end with just a 'stirring effect.' Because the milk moves well, there is always a temperature difference between the glass surface of the bottle and the milk, so the heat exchange also accelerates. In other words, in addition to mixing uniformly, improving heat transfer is the reason why cooling speeds up all at once.

✔︎ Expressing heat transfer with a formula

If we let $${Q}$$ be the amount of heat transfer per unit time, that $${Q}$$ can be simplified into a formula using the following general equation.

$$
Q = U・A・\Delta T
$$

Here

• $${U}$$: Heat transfer coefficient (a value determined by basic experiments)

• $${A}$$: Area through which heat is transferred

• $${\Delta T}$$: Temperature difference

This time, the heat transfer path is milk → bottle → running water, so if we use this general equation to formulate the amount of heat transfer for each path, it looks like this.

▶ Milk → Bottle inner wall

$$
Q_1 = h_{\text{milk}}・A・ \Delta T_{\text{milk-bottle inner}}
$$

▶ Bottle inner wall → Bottle outer wall
　 (Conduction inside the bottle)

$$
Q_2 = \frac{\lambda}{\delta} A \Delta T_{\text{inside-outside bottle}}
$$

▶ Bottle outer wall → Running water

$$
Q_3 = h_{\text{water}}・A・\Delta T_{\text{outside bottle-water}}
$$

Here

• $${h_{\text{milk}}}$$: Convective heat transfer coefficient on the milk side

• $${h_{\text{water}}}$$: Convective heat transfer coefficient on the water side

• $${\lambda}$$: Thermal conductivity of the bottle

• $${\delta}$$: Thickness of the bottle

• $${A}$$: Surface area through which the baby bottle transfers heat (inner wall ≒ outer wall)

• $${\Delta T_{\text{milk-inside bottle}}}$$: Temperature difference between the milk and the inner wall of the bottle

• $${\Delta T_{\text{inside bottle-outside bottle}}}$$: Temperature difference between the inner and outer walls of the bottle

• $${\Delta T_{\text{outside bottle-water}}}$$: Temperature difference between the outer wall of the bottle and the water

What is important here is that $${Q_1=Q_2=Q_3}$$.

Heat is transferred continuously from the milk to the running water. Heat does not escape or increase along the path. Therefore, if heat transfer is slow at any point in the path, that becomes a bottleneck and limits the overall amount of heat transfer.

✔︎ Two effects that occur when you rotate the baby bottle

If you leave the baby bottle as it is, there will be significant unevenness in how it cools. Even if the area near the glass surface of the baby bottle becomes lukewarm, the center remains hot, and the whole thing does not cool down easily. This is the reason why you feel it 'does not cool down easily' when left alone.

On the other hand, when you spin the baby bottle, the following two effects work simultaneously.

1. Effect of eliminating temperature unevenness
   As the entire milk inside moves well, the hot and cold parts mix. As a result, the entire milk begins to cool down all at once.

2. Effect of increasing the convective heat transfer coefficient
   When the milk moves, the flow velocity difference near the bottle wall (glass surface) increases. When expressing heat transfer with the formula mentioned earlier, it is experimentally known that the larger the flow velocity difference near the wall surface, the larger the convective heat transfer coefficient $${h}$$. Therefore, as a result, the speed at which heat moves also increases. You cannot change the material (thermal conductivity $${{\lambda}}$$ ) or thickness $${{\delta}}$$ of the bottle, but you can create the flow velocity difference near the wall surface with your own hands. In other words, the action of 'spinning the baby bottle quickly' is the only parameter you can work on to increase the cooling speed.
   

✔︎ Physics hidden in everyday actions

When cooling baby formula, the speed at which it cools changes significantly depending on whether you rotate the baby bottle. The speed also varies based on 'how hard you try to rotate it.' When you're doing a middle-of-the-night feeding and thinking, 'I need this to cool down even a second faster,' this knowledge makes a big difference.

What you gain by rotating the baby bottle is not just a 'mixing effect' that eliminates temperature unevenness in the milk. You also gain the effect of increasing the convective heat transfer coefficient by increasing the flow velocity difference near the bottle wall. These dual effects accelerate cooling all at once.

Even if you can't change the material or thickness of the bottle, anyone can use the trick of spinning it around by hand. This was one example of how physics knowledge is applied to childcare.

'What I learned in school is useful in daily life.'
Just thinking that might make middle-of-the-night feedings a little bit easier... actually, it probably won't. I'm sorry.🙇

In any case, I am praying that it doesn't turn into a living hell.😊👍

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• Mathematical theme: #ConvectiveHeatTransferCoefficient

• Subject: #BabyBottle #MilkCooling

• Usage scene: #IndependentResearch #Relearning #Reviewing #ClassroomMaterial #Physics

• Topic: #ILovePhysics #LearnFromThisExperience