Mr. Bubble Was Confused. A Cliffhanger.

This year we experienced a record-breaking January in Austria – the coldest since 30 years. Our heat pump system produced 14m3 of ice in the underground tank.

The volume of ice is measured by Mr. Bubble, the winner of The Ultimate Level Sensor Casting Show run by the Chief Engineer last year:

The classic, analog level sensor was very robust and simple, but required continuous human intervention:

Level sensor: The old way

So a multitude of prototypes had been evaluated …

Level sensors: The precursors

The challenge was to measure small changes in level as 1 mm corresponds to about 0,15 m3 of ice.

Mr. Bubble uses a flow of bubbling air in a tube; the measured pressure increases linearly with the distance of the liquid level from the nozzle:

blubber-messrohr-3

Mr. Bubble is fine and sane, as long as ice is growing monotonously: Ice grows from the heat exchanger tubes into the water, and the heat exchanger does not float due to buoyancy, as it is attached to the supporting construction. The design makes sure that not-yet-frozen water can always ‘escape’ to higher levels to make room for growing ice. Finally Mr. Bubble lives inside a hollow cylinder of water inside a block of ice. As long as all the ice is covered by water, Mr. Bubble’s calculation is correct.

But when ambient temperature rises and the collector harvests more energy then needed by the heat pump, melting starts at the heat exchanger tubes. The density of ice is smaller than that of water, so the water level in Mr. Bubble’s hollow cylinder is below the surface level of ice:

Mr. Bubble is utterly confused and literally driven over the edge – having to deal with this cliff of ice:

When ice is melted, the surface level inside the hollow cylinder drops quickly as the diameter of the cylinder is much smaller than the width of the tank. So the alleged volume of ice perceived by Mr. Bubble seems to drop extremely fast and out of proportion: 1m3 of ice is equivalent to 93kWh of energy – the energy our heat pump would need on an extremely cold day. On an ice melting day, the heat pump needs much less, so a drop of more than 1m3 per day is an artefact.

As long as there are ice castles on the surface, Mr. Bubble keeps underestimating the volume of ice. When it gets colder, ice grows again, and its growth is then overestimated via the same effect. Mr. Bubble amplifies the oscillations in growing and shrinking of ice.

In the final stages of melting a slab-with-a-hole-like structure ‘mounted’ above the water surface remains. The actual level of water is lower than it was before the ice period. This is reflected in the raw data – the distance measured. The volume of ice output is calibrated not to show negative values, but the underlying measurement data do:

Only when finally all ice has been melted – slowly and via thermal contact with air – then the water level is back to normal.

In the final stages of melting parts of the suspended slab of ice may break off and then floating small icebergs can confuse Mr. Bubble, too:

So how can we picture the true evolution of ice during melting? I am simulating the volume of ice, based on our measurements of air temperature. To be detailed in a future post – this is my cliffhanger!

A Sublime Transition

Don’t expect anything philosophical or career-change-related. I am talking about water and its phase transition to ice because …

…the fact that a process so common and important as water freezing is not fully resolved and understood, is astonishing.

(Source)

There are more spectacular ways of triggering this transition than just letting a tank of water cool down slowly: Following last winter’s viral trend, fearless mavericks turned boiling water vapor into snow flakes. Simply sublime desublimation?

Here is an elegant demo of Boiling water freezing in midair in the cold:

The science experiment took its toll: About 50 hobbyist scientists scalded themselves, ignoring the empirical rule about spraying any kind of liquid and wind direction:

“I accidentally threw all the BOILING water against the wind and burnt myself.”

Can it really be desublimation of water vapor? The reverse of this process, sublimation, is well known to science fiction fans:

Special effects supervisor Alex Weldon was charged with devising a way to realistically recreate the look of pools of steaming milky water that had been at the location. He concocted similar liquid with evaporated milk and white poster paint, mixed with water and poured into the set’s pools. Steam bubbling to the top was created with dry ice and steam machines, passed into the water via hidden tubing.

(Source: Star Trek online encyclopedia Memory Alpha on planet Vulcan.)

Dry ice is solid carbon dioxide, and it is the combination of temperature and atmospheric pressure on planet earth that allow for the sublimation of CO2. The phase diagram shows that at an air pressure of 1 bar and room temperature (about 293 K = 20°C) only solid and gaseous CO2 can exist:

Carbon dioxide p-T phase diagramIf a chunk of dry ice is taken out of the refrigerator and thrown onto the disco’s dance floor it will heat up a bit, and cross the line between the solid and gas areas in the diagram.

Sublimation of dry ice (Wikimedia, public domain)On the contrary, the phase diagram of water shows that at 1 bar (= 100 kPa) the direct transition from vapor to ice is the is not an option. Following the red horizontal 1-bar-line you need to cross the green realm of the liquid phase:

Phase diagram of water (Wikimedia, User cmglee)You would need to do the experiment in an atmosphere less than 1/100 as dense to sublimate ice or desublimate vapor.

But experiments show that the green area seems to be traversed in the fraction of a second – and boiling water seems to cool down much faster than colder water!

It seems paradoxical as more heat energy need to be removed from boiling water (or vapor!) to cool it down to 0°C. The heat of vaporization is about 2.300 kJ/kg whereas the specific heat of water is only 4 kJ/kgK.

I believe that the sudden freezing  is due to the much more efficient heat transfer between the ambient air and vapor / tiny droplets versus the smaller heat flow from larger droplets to the air.

Mixing water vapor with air will provide for the best exposure of the wildly shaking water molecules to the slower air molecules. If not-yet-vaporized water droplets are thrown into the air, I blame the faster freezing on water’s surface tension decreasing with increasing temperature:

Temperature dependence surface tension of waterSurface tension indicates the work it takes to create or maintain a surface between different phases or substances. The internal pressure inside a water droplet is proportional to surface tension and inverse proportional to its radius. This follows from the work against air pressure needed to increase the size of a droplet. Assuming that droplets of different sizes will be created with similar internal pressures, the average size of droplets will be smaller for higher temperatures.

A cup of water at 90°C will be dispersed into a larger number of smaller droplets and thus a bigger surface exposed to air than a cup at 70°C. The liquid with the lower surface tension will evaporate more quickly.

One more twist: If droplets are created in mid air, as precipitates from condensation or desublimation, it takes work to create their surfaces – proportional to surface tension and area. On the other hand, you gain energy from  these processes – proportional to volume. If the surface tension is lower but the area is larger the total volume is the same – and thus the net effect in terms of energy balance might be the same. But arguments based on energy balance only don’t take into account the dynamic nature of this process, far off thermodynamic equilibrium: The theoretical energy gain can only be cashed in (within the time frame we are interested in it) if condensation or freezing or desublimation is actually initiated – which in turn depends in the shape and area of the surface and on nuclei for droplets.

Heat transfer is of course more efficient for a larger temperature differences between air and water; perhaps that’s why the trend started in Siberia:

I have for sure not discussed any phenomenon involved here. Even hot water kept in a vessel can cool down and freeze faster than initially cooler water: This is called the Mpemba effect, a phenomenon known to our ancestors and rediscovered by the scientific community in the 1960s – after a curious African student refused to believe that his teachers called his observations on making ice cream ‘impossible’. The effect is surprisingly difficult to explain!

In 2013 an Mpemba effect contest had been held and the paper quoted at the top of this post was the winner (out of 22.000 submissions!). Physical chemist Nikola Bregovic emphasizes the impact of heat transfer and convection: Hot water is cooled faster due to more efficient heat transfer to the environment. Stirring the liquid will disturb convective flows inside the vessel and can prevent the Mpemba effect.

The  effect could also be due to different spontaneous freezing temperatures of supercooled water. Ice crystals can start to grow instantly at a temperature below the theoretical freezing point:

Various parameters and processes – such as living organisms in the water or heating water to higher temperatures before! –  might destroy or create nucleation sites for ice crystals. Supercooling of vapor might also allow for a jump over the green liquid area in the phase diagram, and thus for deposition of ice from vapor even at normal pressures.

Quoting Bregovic again:

I did not expect to find that water could behave in such a different manner under so similar conditions. Once again this small, simple molecule amazes and intrigues us with it’s magic.
~
Ice in our underground water tank, growing at the top layer of heat exchanger tubes. These are only covered with water if a bulk of ice underneath will make the water level rise.

Non-Linear Art. (Should Actually Be: Random Thoughts on Fluid Dynamics)

In my favorite ancient classical mechanics textbook I found an unexpected statement. I think 1960s textbooks weren’t expected to be garnished with geek humor or philosophical references as much as seems to be the default today – therefore Feynman’s books were so refreshing.

Natural phenomena featured by visual artists are typically those described by non-linear differential equations . Those equations allow for the playful interactions of clouds and water waves of ever changing shapes.

So fluid dynamics is more appealing to the artist than boring electromagnetic waves.

Grimshaw, John Atkinson - In Peril - 1879

Is there an easy way to explain this without too much math? Most likely not but I try anyway.

I try to zoom in on a small piece of material, an incredibly small cube of water in a flow at a certain point of time. I imagine this cube as decorated by color. This cube will change its shape quickly and turn into some irregular shape – there are forces pulling and pushing – e.g. gravity.

This transformation is governed by two principles:

  • First, mass cannot vanish. This is classical physics, no need to consider the generation of new particles from the energy of collisions. Mass is conserved locally, that is if some material suddenly shows up at some point in space, it had to have been travelling to that point from adjacent places.
  • Second, Newton’s law is at play: Forces are equal to a change momentum. If we know the force acting at time t and point (x,y,z), we know how much momentum will change in a short period of time.

Typically any course in classical mechanics starts from point particles such as cannon balls or planets – masses that happen to be concentrated in a single point in space. Knowing the force at a point of time at the position of the ball we know the acceleration and we can calculate the velocity in the next moment of time.

This also holds for our colored little cube of fluid – but we usually don’t follow decorated lumps of mass individually. The behavior of the fluid is described perfectly if we know the mass density and the velocity at any point of time and space. Think little arrows attached to each point in space, probably changing with time, too.

Aerodynamics of model car

Digesting that difference between a particle’s trajectory and an anonymous velocity field is a big conceptual leap in my point of view. Sometimes I wonder if it would be better to not learn about the point approach in the first place because it is so hard to unlearn later. Point particle mechanics is included as a special case in fluid mechanics – the flowing cannon ball is represented by a field that has a non-zero value only at positions equivalent to the trajectory. Using the field-style description we would say that part of the cannon ball vanishes behind it and re-appears “before” it, along the trajectory.

Pushing the cube also moves it to another place where the velocity field differs. Properties of that very decorated little cube can change at the spot where it is – this is called an explicit dependence on time. But it can also change indirectly because parts of it are moved with the flow. It changes with time due to moving in space over a certain distance. That distance is again governed by the velocity – distance is velocity times period of time.

Thus for one spatial dimension the change of velocity dv associated with dt elapsed is also related to a spatial shift dx = vdt. Starting from a mean velocity of our decorated cube v(x,t) we end up with v(x + vdt, t+dt) after dt has elapsed and the cube has been moved by vdt. For the cannon ball we could have described this simply as v(t + dt) as v was not a field.

And this is where non-linearity sneaks in: The indirect contribution via moving with the flow, also called convective acceleration, is quadratic in v – the spatial change of v is multiplied by v again. If you then allow for friction you get even more nasty non-linearities in the parts of the Navier-Stokes equations describing the forces.

My point here is that even if we neglect dissipation (describing what is called dry water tongue-in-cheek) there is already non-linearity. The canonical example for wavy motions – water waves – is actually rather difficult to describe due to that, and you need to resort to considering small fluctuations of the water surface even if you start from the simplest assumptions.

The tube