Energy used to boil a full kettle once vs boiling enough water for a cup multiple times

Question

I have an electric kettle and a thermal carafe. The carafe itself is not heated, it's basically just a large thermal flask. I've noticed that when hot water is kept in the carafe, the water stays hot for a long time and so I was wondering whether it was more energy efficient to boil a full kettle and store the water in the carafe, using it for several hot drinks over the course of a few hours, or if it is more energy efficient to boil enough water for single drink multiple times over the course of the day.

Let's say a full kettle makes 6 mugs of hot drink so it's between boiling 6 mugs' worth of water once, or 1 mug's worth of water 6 times.

Answer 1

So, based on the information given by OP in the comments, ideally there should be no difference. This is because the energy required to heat a substance of mass $m$ by temperature $\Delta T$ is $mC\Delta T$, where $C$ is the specific heat. Hence $$mC\Delta T=\sum_i(m_iC\Delta T)$$ Thus heating separately in bulk or in parts should not matter.

Practically however there will be a difference in efficiency. The primary couple of reasons that I think are relevant are

a) Some amount of heat is wasted on the kettle. Let an amount $m$ of water be present in the kettle and let $E(m)$ denote the amount of energy wasted on the kettle to bring the water to a boil. In this case the total amount of energy used is $E(m)+mC\Delta T$. Now suppose each of your cups use mass $m$ and in a day you use $n$ such cups, then you have to compare $nE(m)+nmC\Delta T$ and $E(nm)+nmC\Delta T$ i.e in effect compare $nE(m)$ and $E(nm)$.
Now my guess is that the kettle will have been built for peak efficiency with some volume in mind which will probably be leaning towards it being full. Thus I feel practical purposes $$E(nm)<nE(m)$$ You can probably experiment to find out whether this holds true or not.

b) As OP states they keep the kettle on till it switches off. This implies some amount of time is spent at the end where the kettle is on but no energy is spent on increasing the temperature of the water. Its a fairly good approximation to treat this overtime as a constant wrt the amount of water. Thus in the case of smaller amounts of water more time in total is spent in this overtime stage again leading to wastage of energy.

In summary according to the above elaborations and assumptions it is more efficient to heat the water for the day and use it as and when required.

Answer 2

The energy should be equal at first sight, but there are a few things to consider.

Most kettles continue heating slightly longer than necessary before they switch off, so by using the flask this wasted energy could be reduced.

There is the energy needed to heat the kettle itself to consider, also kettles have a minimum mark, so maybe you are boiling slightly more than you need each time.

On the other hand - if you decide only to have 5 drinks during the day, instead of 6, then making separate drinks maybe better.

As you'll know the details, perhaps you could consider the above when deciding.

Assuming the kettle delivers the energy at a constant rate, i.e. the power is constant, you could find out experimentally by doing a few days using the flask and time how long the kettle is on for, then a few days making drinks as you need them, time that too.

Try and pick days where the temperature is similar outside. The method that uses the least time would be most energy efficient.