Willis Heating System

I guess it depends upon what one regards as "noticeably cooler" - under normal circumstances (i.e. unless we have 'a house full of visitors') the water late at night is 'well hot enough' for anything (well, I can't say about baths, since we don't really do them!)
Anyway, I'm acquiring some 'Chapter and Verse' for you all today. I measured the temp at the kitchen tap first thing this morning and, if I remember, will do the same late tonight - so 'watch this space'!
OK, 'this space'; is now largely ready to be watched ....

The water in my cylinder was last heated during some of the period from 03:00 -06:00 'last night' - so around 21-24 hours ago.

The temp of hot water at my kitchen tap first thing in the morning (around 08:00, hence not that long after the heating, was 61.3C. Now, 21+ hours after the last heating, it is 56.4°C - a fall of 4.9°C and still too hot to hold my hand under for any significant period of time.

The point of this exercise was to hopefully confirm that I'm not experiencing significant mixing of stored heated water and the cold water drawn into the cylinder during the day, and I think that these figures have indeed confirmed that.

I would guess that we've probably used about half a cylinder of hot water today (70L), hence will have drawn that much cold water into the bottom of the DHW cylinder but come tomorrow I'll know how much electricity was used by the immersion tonight, hence will be able to make a more accurate estimate of how much cold water was actually sitting below the heated water. I will then be able, for comparison, to estimate what the water temp would have been now if there had their been complete mixing of the heated water with the cold water below it.
 
Exactly, despite what you wrote about your hypothetical water inlet that would use a jet of mains-pressure water as a 'stirrer'
I think you missed the point about having things that strive to diminish or strive to preserve the stratification
Sure, that's true IF one starts with a cylinder completely full of cold water
No, it could be any temp

then switches on the heater and wants only a very small amount of hot water (at 'red' temp). If one wants more than a very small amount of water, with the Willis one will have to wait, for anything up to 50 mins with your figures
Oh good lord. You're completely missing the point, I think deliberately. I genuinely wonder why.

They'll have to wait longer with the immersion, and pay more

i think there will be an amount of temp gradient
My observations differ, because out of idle curiosity I've watched the rise in the 4 temp sensors attached to different heights in my tank, connected to the monitoring software that draws a profile of the tank, as it heats. I understand your picture but I think you'll struggle to recreate it

Conduct the experiment yourself with a pan and a thermometer; see how much of a gradient you don't get in the water between the surface (top of tank) and the bottom (immediately above the heater)

with some conduction warming it further down
No, water is a relatively poor conductor to itself.
I'm convinced the copper cylinder feels gradual variation
Maybe you never owned an uninsulated (or jacketed) cylinder with a coal fire back boiler; as a youngster I did, and you could tell when there was enough hot water for a shower by sticking your hand inside the jacket and feeling down the tank. The temperature change was clearly defined

An element close to the top of a cylinder will provide pretty fast 'replenishment'.
But only for the volume in which it sits, and above it. You'll really struggle to get your "at the top" immersion to heat the full tank. If you want just a bit more than it can heat, you'll need another just a bit further down

It's possibly to buy tanks with immersions; if an immersion would heat and stratify well from the start, the hot stream of water leaving the element, going straight to the top of the tank, fanning out and sitting there, not passing go, and not collecting 200 cold from the water around it, we wouldn't need two - just fit one at the bottom and it can heat the tank stratified..

So, it doesn't work like that, what the heck, immersions are cheap, fit 10 of them to your tank then you can choose whether to heat 10% of it, or 20%, or 30%...

Or just fit one Willis and run it for longer if you want 10 or 20 or 30...

A Willis is the bartender that mixes the cocktail topping in a separate container, and then carefully pours it over the main drink leaving it stratified from the get go. Immersions aren't so clear cut stratified from the get go
 
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Well I am still mystified as to the answer.
As far as I can see there are three possible answers really.

1/ There is no clear cut advantage overall of the great difference of two immersion heaters in a cylinder or one immersion heater in the cylinder and a Willis Heater outside of the cylinder.

2/ There is a clear cut advantage/disadvantage of one or the other in the gen population.

3/ There might well be some advantages/disadvatages that overall suits certain members of the population according to their mode of normal behaviour much of the time - indeed once that is recognised that those who are prepared to modify their behaviour one way or the other much more of the time then the advantage/disadvantage ratio could be made to change to a significant improvement much of the time (Horses for Courses) even though there is no clear cut one size fits all situation , time and power of charging up the relevant parts and timings a durations of draw off for usage could all play a part.

If we think of a simpler idea of illustration of ideas then the concept of water towers over an area for the water supply of a town/village/hamlet - the means of pumping does not need to equal the maximum duration throughout the day/week/month/year but so long as it is at least equal to the average at all times and never fall below the maximum flow versus storage equation at all times then we can end up with less powerfull less expensive means of pumping, win/win all round.
 

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