**During my mug cooling experiment last week**, I was surprised to find that taking the lid off a vacuum insulated mug increased its initial cooling rate by a factor 7.5.

**Removing the lid allowed air **from the room to flow across the surface of the water, cooling it in two ways.

- Firstly, the air would warm up when it contacted the hot water, and then carry heat away in a convective flow.
- Secondly, some hot water would evaporate into the moving air and carry away so – called ‘latent heat’.

**I wondered** which of these two effects was more important?

**I decided to work out the answer by calculating** how much evaporation would be required to explain ALL the cooling. I could then check my calculation against the measured mass of water that was lost to evaporation.

**Where to start?**

**I started with the cooling curve** from the previous blog.

**Because I knew the mass of water (g) and its heat capacity (joule per gram per °C)**, I could calculate the rate of heat loss in watts required to cool the water at the observed rate.

**In Graph#2 below** I have plotted this versus the difference in temperature between the water and the room temperature, which was around 20 °C.

**I was struck by two things: **

**Firstly**, without the lid, the rate of heat loss was initially 40 watts – which seemed very high.**Secondly:**the rate of heat loss was almost a perfect straight line This is broadly what one expects in a wide range of heat flow problems –**When the lid was on,***the rate of heat flow is proportional to the temperature difference*. But…**When the lid was off**, the heat flow varied non-linearly with temperature difference.

**To find out the effect of the lid**, I subtracted the two curves from each other to get the *difference in heat flow* versus the temperature of the water above ambient (Graph#3).

[* Technical Note: Because the data in Graph#2 is very noisy and irregularly spaced, I used Excel™ to work out a ‘trend line’ that describes the underlying ‘trend’ of the data. I then subtracted the two trend lines from each other.*]

**This curve now told me the extra rate of cooling** caused by removing the lid.

**If this was ALL due to evaporative cooling**, then I could work out the expected loss of mass by dividing by the latent heat of vaporisation of water (approximately 2260 joules per gram) (Graph#4).

**Graph#4 told me the rate** at which water would need to evaporate to explain ALL the cooling caused by removing the lid.

**Combining that result with the data in Graph#1, **I worked out the cumulative amount of water that would need to evaporate to explain ALL the observed extra cooling (Graph#5)

**In Lid-Off Experiments#1 and #2, **I had weighed the water before and after the cooling experiment and so I knew that in each experiment with the lid off I had lost respectively 25 g and 31 g of water – just under 10% of the water.

**But Graph #5 really needed some data** on the *rate* of mass loss, so I did an additional experiment where I didn’t measure the temperature, but instead just weighed the mug every few minutes. This is the data plotted on Graph#5 as discrete points.

**Conclusions#1**

**In Graph#5, it’s clear that the measured rate of evaporation** can’t explain *all* the increased cooling rate loss, but it can explain ‘*about a third of it*‘.

**So evaporation is responsible for about** a third of the extra cooling, with two thirds being driven by heat transfer to the flowing air above the cup.

**It is also interesting that even** **though** the cooling curves in Graph#1 are very similar, the amount of evaporation in Graph#5 is quite variable.

**The video below is backlit** to show the ‘steam’ rising above the mug, and it is clear that the particular patterns of air flow are very variable.

**The actual amount of evaporation depends on the rate of air flow across the water surface, **and that is driven both by

**natural convection**– driven by the hot low-density air rising, but also by…**forced convection**– draughts flowing above the cup.

**I don’t know, but I suspect it is this variability** in air flow that caused the variability in the amount of evaporation.

**Conclusions#2**

**I have wasted spent a several hours on these calculations. **And I don’t really know why.

**Partly, I was just curious** about the answer.

**Partly, I wanted to share my view** that it is simply amazing how much subtle physics is taking place around us all the time.

**And partly, I am still trying to catch my breath** after deciding to go ‘part-time’ from next year. Writing blog articles such as this is part of just keeping on keeping on until something about the future becomes clearer.

**P.S. Expensive Mugs**

**Finally, on the off-chance that** (a) anybody is still reading and (b) they actually care passionately about the temperature of their beverages, and (c) they are prepared to spend £80 on a mug, then the Ember temperature-controlled Ceramic mug may be just thing for you. Enjoy 🙂