Released from ice are brook and river
By the quickening glance of the gracious Spring;
The colors of hope to the valley cling,
And weak old Winter himself must shiver,
Withdrawn to the mountains, a crownless king.
These are the first lines of the English version of a famous German poem on spring, from the drama Faust, by Johann Wolfgang von Goethe. Weird factoid about me: I was once inclined to study literature, rather than physics. But finally physics won, so this is a post about joyful toying with modeling heat transport in ice and water.
After 46 days we had a high score: The ice cube, generated by our heat pump, stopped growing at about 15m3. About 10m3 of water remained unfrozen. After the volume of ice had been in a steady state for a a while, we turned on the solar collector again to return to standard operations.
Where did the energy for the heat pump come from before?
The lid of the tank is insulated against ambient air, the solar collector was not operational, and no ice had been created: The remaining energy has to be provided by the 5th element that cannot be shut off: 1) water 2) ice, 3) ambient air, 4) solar radiation … 5) ground.
Normally ground supplies about 15 W per m2 surface area – deduced from monitoring the power transported with the brine flow and energy accounting for the tank. The active interface between tank and ground below frost depth is about 35 m2. This results in about 0,5 kW in total, thus just 12 kWh per day, much lower than the ~ 50 kWh ambient energy fed into the heat pump.
After much deliberation and playing with the heat transfer equation we came up with this description of the evolution of the ice cube:
Phase 1: Growth of ice into water.
- Ice starts to grow from the heat exchanger tubes into the remaining water. These tubes are installed in a meandering pattern, traversing the storage tank.
- At some point the thick layers of ice covering adjacent parts of the pipes touch each other. The surface of this solid ice cube is smaller than the interface between the meandering ice formations and water before. The power needed by the heat pump has to be pushed through a smaller surface – which is only possible if the temperature gradient within the ice gets larger. As the temperature at the ice-water interface has to be 0°C, the temperature at the heat exchanger has to decrease. This is exactly what we see from monitoring data – brine temperature drops well below 0°C.
- Side-effect: Due to the lower brine temperature the coefficient of performance decreases slightly. So more of the total heating energy needs to be provided by the electrical input. We call this the heat source paradox: The worse performance is, the more you spare the energy stored in the heat source. Thanks to this self-protection mechanism, the energy in the tank will not suddenly drop to zero.
Phase 2: Ice touching ground.
- As long as there is some water between ice and ground, the water temperature is 0°C. This is the temperature ground ‘sees’ and the temperature which is relevant for the low heat transport from ground to water.
- Ice touches some surfaces of the cuboid tank – the ones where the heat exchanger tubes are closest to the surface. Now ground is directly connected to ice with its temperatures < 0°C. The temperature gradient between ground and ice provides for a higher flow of energy. This is also indicated by the evolution of the temperature in the ground below the tank: While temperatures of undisturbed ground and the region below the tank had been aligned before, ground temperature beneath the tank still kept getting lower – although a few meters away from the tank ground is already warming up again.
- If enough heat is delivered by ground, no more heat is needed by freezing the remaining water in the tank. When ground temperature reaches zero, it can even freeze – which happens with geothermal systems, too. We might have extended the ice storage into ground.
Heat transport within ice is actually more efficient than transport in water: Ice has 4 times the heat conductivity of water, and 10 times the thermal diffusivity. The latter is a measure for the time a deposited ‘lump of heat’ will be spread in space.
So we have built a very efficient cold bridge between the heat exchanger and ground. Everything is consistent with the poetry of the differential equation of heat transfer.
I marvel at the intriguing and mathematically appealing physics in my backyard!
elkemental Force 
It will be interesting to see what happens next. The ice has to absorb an enormous quantity of heat now if you expect all of it to melt over the next yearly cycle. This is quite a different mode of operations that I would have expected since i initially figured you were just relying on the extraction of heat present during just one physical phase–liquid. Until you brought it up I never would have even dreamed that removing Hf would have been feasible at all. You only indicated a degradation in performance. Me–I would have expected more-or-less a drop to 1:1 in the system!
During my days as a physics teacher I always said to others that never would a month go by but I would discover, just through the routine of doing my job, something fundamentally new. And so it continues :-)
But what about physical damage to the interior pipes and tank liner? Did you observe anything?
As the plot shows we are down to about 5m3 again :-) The solar collector has melted 2/3 of the ice already – within about 10 days! Not only is convection rather effective, but the ice in the tank even helps as the temperature difference between ambient air and the tank is kept at a high level. We could have kept the ice longer to support passive cooling in early summer but we also wanted to demonstrate how effective the solar collector is even when ambient temperature is still low.
The performance factor just dropped a bit below 4 (usually it is higher) when the source temperature dropped as the ice itself is cooled – the minimum brine temperature was -5°C. But as long as the heat source is still abile to deliver the required flow of heat performance should not drop dramatically … just by a factor dictated by Carnot’s coefficient of performance … thus as long as there is some water to the frozen, within the tank or in the surrounding ground. I would expect another drop in brine temperature after a that period of steady state, when ground has been exploited too much (That’s what you encounter with geothermal systems).
Before the enourmous heat source would be completely “used up” you would rather turn on the heating element triggered by a minimum temperature – then you would heat 1:1 (About once in 10 years for a day or so).
We don’t see any damage, and monitoring of pressure does not indicate any leaks. The structures in the melting ice underwater were interesting as far as we see or rather touch them – we should install underwater cameras someday.
Thank you for making me pick up my copy of Faust :-)
I also re-read it from time to time – the density of ‘lines which became proverbs’ is amazing.
I might have hust had a sort of epiphany: when you mentioned that you were torn between literature and physics, I was reminded of my Abiturklasse, in which I focused on Literature and Physics – I was nudged away from literature by my dogmatic teacher. But this conflict seems to be present in many scientists – consider the characters from the Big Bang Theory: Lennard, Sheldon, Raj, and Howard are accomplished Physicists – and an Engineer – yet their passion for graphic novels rivals their passion in academia. An intellectual counterweight, if you will
Another combination people pointed out to me is music and physics – like Einstein playing the violin …
I imagined a ‘scientist’ being a Leonardo-da-Vinci-style erudite renaissance person so it did not feel like a conflict.
Well, I got Music, Literature and Physics. I was certainly torn between the three of them as a career, since having two of the three wouldn’t have been difficult enough
I’m sorry, I’m still staggering at the thought of fifteen cubic meters of ice. Wow.
Absolutely – and I would have laughed to see it. We should have installed underwater lighting and cameras. Briefly we had considered to drain the water just to see the ice cube but decided otherwise as it would have interfered with the experiment to much.
We must have hit the end of the comments layers, or whatever that’s called that limits how far we can exchange… I didn’t even see your response in my comments notification. So, starting a new thread: I’ve also seen documentaries with different types of composting tree trunks that are used for growing mushrooms. I once saw in a local gardener’s catalogue a ‘mushroom kit’ that people can buy for DIY fungal horticulture. I wonder if such a kit would be a good place to start?
Yes, I have seen such kits, too – a box with special soil (?) and the spores (?) … or whatever :-)
But before my goal is to use the garden ‘more efficiently’. The ‘flowers’ I planted (or at least I did not weed out) were not the most beautiful ones but rather those that will survive on zero extra watering – so I plan to replace some of the ‘decorative’ stuff by edible plants… some of which actually look quite nice in my opinion. From photos I’d say that also eggplants have their aesthetic merits :-)
Thanks for throwing a few literary breadcrumbs in here. I didn’t want to jump in until the science enthusiasts went back to work; I’d merely clog the exchange with an off-topic question like, are those tomatoes or eggplants in the seedling boxes? Also, I like the addition of the About Me tagline, “I build stuff that works” on your home page. Completely to the point. :)
:-) You’re an expert, Michelle – these are eggplants, but we have lots of tomato seedlings elsewhere in the office. I don’t expect too much as the recommendation for eggplants in our climate is: You can try outside a greenhouse but only if wine grows in your region – but the fruits will be much smaller. I live in ‘Austria’s oldest winemaking village’, so we will try :-)
My about.me tagline is what I might change my LinkedIn tagline into someday :-)
I’m not wise in the ways of identifying seedlings, but I could remember your plans for the spring. I hope the indoor headstart helps. They look very healthy.
I revised my linkedin a while ago and finally finished the basic profile. I honestly don’t enjoy the platform. I think I’m passing into a creative phase right now where I’d rather be only making things, and am really struggling with the business side. This probably reflects a recent imbalance the other way. LinkedIn leaves me feeling guilty.
I try to keep to my increasing-success-by-lowering-expectations attitude :-) Every eggplant survivor carrying one small fruit will be a success :-) Yes, I wanted to try that for a while – thanks for remembering! My other secret dream is to grow mushrooms in our second earth cellar, the one one beneath the house which we haven’t used for energy storage yet.
That would indeed be interesting. I think there’s an art to mushroom farming, from the little I know.
I have seen some DIY solutions for mushrooms; I was intrigued by placing straw in a laundry basket so that mushrooms would grow through the holes at the sides. It looked so simple, but I guess it takes a lot of trials.
Knowing absolutely nothing about thermodynamics I was intrigued by the carnotcycle blog (he found mine!) . Result so far – I have some clue as to what you are writing about! here is the latest carnot post:
https://carnotcycle.wordpress.com/2015/04/01/spontaneity-and-the-performance-of-work/
Thanks, Howard – I have been following carnotcycle for a long time but his latest post is yet awaiting my attention ;-)
As you probably know, I was an operator at CERN, at the Low Energy Antiproton Ring (LEAR) specifically. This ring carried a bunch of antiprotons in orbit, maintained with magnets. The particle bunch had a distribtion – typically gaussian. When you applied an RF noise over a given bandwidth, a diffusion process kicked in and the particle distribution began to flatten. With the RF power and the bandwidth you could control which part of the distribution was affected and how fast.
So,extraordinary as it may sound, having no other tools at my disposal: I studied diffusion with antiprotons!!!
An old paper I co-authored here: https://accelconf.web.cern.ch/accelconf/e94/PDF/EPAC1994_2376.PDF
(I have since left science :( )
When Joseph Fourier formulated the heat equation (no referred to diffusion equation as you know), he had no tools to solve it. He went ahead and developed what is now known as Fourier Analysis and decomposed the initial distribution into a sum of harmonics. Since the heat equation is easily solved for a sine or an exponential (as in Green’s I believe), he had cracked the problem. He had no inkling of the impact of his harmonic analysis tool would have!!!
I love that you did the numerical analysis of the your melting ice. Having played with and observed diffusion for years, I have an intuitive understanding of the mechanism and can sketch the evolution of any distribution exposed to diffusion with the need for calculations. The process is such that I still believe that a closed solution for any analytically defined distribution must exist without the need for the typical summations series that traditional methods lead to. When I find it, I shall write another paper :)
This is very cool – studying diffusion with antiprotons :-) I think finding analogies and close similarities in underlying theories is one of the most intriguing things in physics. In one of his physics lectures, Feynman demonstrates this when comparing solutions of the heat transport equations to electrical fields (solutions of Maxwell’s equations for electrostatic cases) and the shape of rubber membranes under stress. He tried to make the point that as a physicist you could always contribute something to seemingly unrelated fields because of that. I agree on principle, but I am not sure if this is still valid today (sadly), especially from employers’ perspective – when there is specialized visual software for anything.
Yes, I think using Green’s functions is basically the same you find it via Fourier transforming the heat source term, when solving the differential equation for a delta function shaped inhomogeneous term. In the simple example of an ordinary differential equation for a damped harmonic oscillator Green’s function is a damped oscillation.
I knew Fourier analysis, but I did not know that the heat transport equation was his starting point. It seems I learned some ‘theorems’ without the name of the developer attached (… recently realized that re Noether’s theorem…)…. due to whatever cultural reason. A lack of historical perspective perhaps?
I’d love to know more about the history of these theories. There is a fascinating human story behind everyone of them. Just imagine the correspondence between Einstein and Gödel or Schrödinger!
The story is that Fourier considered Fourier Analysis a sideshow to his heat diffusion research. It was all about steam in those days! No so far from ice storage!
How I agree on the poetry of the heat equation. For years I played around with that equation with numerical solutions because the maths of more sophisticated solutions – other than Fourrier’s own stunningly beautiful solution – was beyond me. I haven’t given up hope to find a closed solution one day for all cases. Probably someone has already proven that there is no such solution or perhaps it already exists but I am not deterred: ignorance has its own poetry.
I am all for analytical solutions – perhaps because it had once (at the university, when working in on PhD) impressed me much how Russian scientists tried to avoid any computation at all costs and go pen and paper all the way … I guessed as they did not have such computational power at that time. What I like about this – and what I found the most intriguing thing in theoretical physics – was to learn something about solutions of equations without actually solving them, e.g. by checking limiting cases that describe the most interesting features of solutions.
Using the Green’s function (representing the ‘response’ to a point-shaped lump of heat) is a an analytic solution in my opinion, but you still need to multiply it with the realistic heat source density and integrate it. I don’t know if this is Fourier’s solution as I never learned about that equation having a specific name tag … after all it is the same differential equation as for diffusion of anything, if local conservation of some quantity is observed (energy, mass).
But since the heat source or the boundary conditions would be described with a messy function in real life anyway – so I think using the Green’s function might create some false sense of solving analytically as the integration has to be done numerically anyway.
Modelling that heat transfer in ground I use a numerical approach right from the start… as all the other stuff to be simulated (valves opening and closing etc.) is described best by what happens in each time slot.
What I tried in this case analytically was to cross-check if the interface between water and ice might slow down in a simplified scenario, just involving an infinite volume of water with a slab-shaped heat exchanger in the middle of it…. and checking the limiting case when the slopes of temperatures distributions are nearly straight, ‘immediately’ before steady state – to check if I can explain the phenomenon at least in part, without taking into account this special interface with ground. Turns out, yes it does (though at much greater distances and so perhaps not relevant for this experiment), as more and more energy is harvested from cooling the ice itself, in comparison to creating new ice. But only after having written several pages I noticed that you could have told from the simplest of energy balances :-)