Pump and Circumstance

My parents are in the process of installing a set of heat pumps in my childhood home, so they don't have to rely on the old wood stove, which requires buying, stacking, and splitting large quantities of lumber every year. I was interested to learn more about these real-world pumps, since physicists are introduced to the idea of heat pumps and heat engines in any intro thermodynamics class.

The idea with both these devices is that we have some working gas that we move to different levels of temperature (T), pressure (p), volume (V), and entropy (S). The reservoir where we keep this gas is attached to a piston/shaft. The piston controls the pressure and volume, and we can either turn the shaft to create a temperature difference (heat pump), or apply a temperature difference to make the shaft turn (heat engine).

Usually these different stages the pump goes through are shown on a p-V diagram, which shows the relationship between pressure and volume at each step. Here's a simple example:

The steps are

1. At a low, constant pressure, expand the gas to a larger volume
2. Keeping that volume fixed, increase the pressure to a high value
3. Decrease the volume again, but at the new pressure
4. Bring the pressure back to the start, at the original volume

The reason physicists like to represent this in terms of pressure and volume is that it's easy to find the energy:

This says that the energy produced by the system is equal to the area under each curve in the plot above. This is where the arrows come in: pointing to the right means the volume is increasing, and dV is positive, while to the left is negative. How does this relate to temperature though? If we're using an ideal gas (air is close), then these factors are related by

so in the plot above, the upper right corner will be the hottest, while the lower left will be the coolest. Since the arrows are to the left at the higher pressure, we put in more energy than we get out, and this is a heat pump that will make that temperature difference.

What's really interesting about heat pumps is their efficiency: The amount of energy you get out (through heating/cooling), divided by the energy you put in (turning the shaft). In the case of an old fashioned electric heater, you run current through a big piece of metal, which heats up, converting 100% of the input energy into heat. It seems odd that you could do better than 100%, but heat pumps manage it.

The key is that we use ambient energy, which doesn't count against our input. In my parents' case, the pump is taking energy from the cold outside, and moving it to the warm inside, so the efficiency is

where Q is the energy exchanged with each location.

When physicists talk about efficiency, we like to bring in the Carnot cycle, which is the theoretical maximum efficiency. The p-V diagram for this case is a bit funny looking:

For this type of pump, we can replace the energy in the efficiency equation with the indoor and outdoor temperatures, in Kelvin. According to Steve's last NOAA report in 2012, the lowest recorded temperature in Ashfield was 249 K. For an indoor temperature of 293 K (brisk for my taste, but I'm sure he would prefer cooler), the efficiency comes to 666%! Clearly these heat pumps are the work of the devil...

Lean on Me

For some reason, my bus the past week has been especially full, forcing me to stand for most of my commute. I'm always surprised by how difficult it is to keep my balance going around curves, so I thought I'd try putting some hard numbers to the issue.

I've talked about the strange, non-inertial effects of being in a turning vehicle before, but this situation is much simpler. When the bus goes around a turn, it drags my feet along with a centripetal acceleration. Balance is all about keeping the total forces on you over your feet:

Here, ar is the centripetal acceleration of the bus, and g is the acceleration from gravity. Using geometry,

When something moves in a circle with radius r and velocity v, the centripetal acceleration is

Putting these together and solving for v gives

The turning radius for a standard city bus is about 21.5 feet, and the maximum angle a person can lean is around 15°. Putting these together gives a speed of just 9 mph! As a non-driver, I don't really have a feel for typical turning speeds, but the National Association of City Transportation Officials suggest a maximum of 15 mph, so our limit is well below this. Of course, when you're riding a bus you can lean into the curve, and hold on to the railings, neither of which we've accounted for here. Still, I feel a little justified in lurching around awkwardly during my commute.

Harry Nyquist the Scientist (Harry! Harry! Harry!)

Ok, doesn't quite scan, so I guess he's not destined for TV stardom, but one of his ideas is at the core of this week's post. Last week, I was talking to my parents over Facetime and Steve, ever the stickler for visual quality, complained that my lights appeared to be flickering:

Looking at it from my end though, the light appeared fine. I realized that the LED bulb must be flickering (by design) at a rate higher than my eyes can perceive, but my laptop camera is recording below the Nyquist rate, causing the flickers to appear.

Before getting into that, let's talk about how LED bulbs work. LEDs require direct current (DC), while house power is alternating current (AC), so the first thing we need to do is convert between these two. Alternating current is a sinusoid, oscillating between positive and negative of a certain value:

There's an electrical component called a diode that only allows current to flow in one direction. In fact, that's the D in LED, but right now we're just talking about the regular kind of diode, not the Light Emitting ones. By arranging a few of those in a clever way, we can switch the negative part of that curve to positive:

Made with circuit-diagram.org

On the left is the AC source, and on the right is a voltmeter. The diodes in the middle only allow current to flow in the direction of the arrow. The voltmeter's output looks something like this:

This isn't ideal though, since the voltage is going all the way down to zero. We need a way to smooth out this curve. We can do that by adding a capacitor, which stores and releases charge as the voltage changes:

Now the voltage looks something like this:

Still not perfect, but it's the best we can do with something simple. Those little bumps mean the LEDs will be flickering a tiny bit, but it's at twice the rate of the original AC input. In France, this is 2x50 times per second (Hz).

The camera on my laptop captures video at about 30 frames per second. According to Nyquist, to accurately record a signal, we need to take data at twice the rate that the data changes, but that would be 200 Hz, far above what my camera is capable of. We can see why this is necessary by considering a point moving in a counter-clockwise circle:

The dot completes a cycle every 8 steps. The image on the left is sampled every step, while the one on the right is sampled every 7 steps. The slower sample rate causes the dot to appear to be moving backwards. You've probably seen a similar effect in videos which include old TVs or computers that used CRT screens. The screen appears to blink because the camera captures the screen at different points in its refresh.

That leaves us with the question of why the light doesn't blink according to our eyes. That's addressed by the flicker fusion threshold, which comes from the spectacularly-named field of psychophysics. Our eyes aren't able to perceive changes in intensity above a certain rate, instead averaging things out. For most people, it's around 80 Hz, so the LEDs easily surpass that, but the captured video is nowhere close.

Thanks for a great question/complaint Steve!