How Thermally Efficient is a Christmas Pyramid?

 

It came without ribbons. It came without tags. It came without packages, boxes or bags. And he puzzled and puzzled ’till his puzzler was sore. Then the Grinch thought of something he hadn’t before. What if Christmas, he thought, doesn’t come from a store. What if Christmas, perhaps, means a little bit more. Doctor Seuss

 

I’ve always been fond of physics-based toys, decorations and gadgets. A few of my favorites are the spinning top, the Chinese drinking bird, Newton’s Cradle, the plasma globe, the radiometer, and, of course, the lava lamp. But perhaps my favorite of all is the Christmas Pyramid. Dating back to the 16th century, the handcrafted tradition of the Christmas pyramid began in the Erzgebirge region, a mountainous and densely coniferous region on the German-Czech border. The pyramids were originally used as a safety device by miners working underground to develop the region’s rich tin, iron and silver deposits. Starting in the 19th century, local woodcarvers evolved the early “light racks” into fanciful home decorations in which rotating propellers are turned by the hot air rising from candles at the base of the pyramid. Weihnachten pyramids are believed to be a precursor of the modern Christmas tree. Just check out the craftsmanship on these well preserved examples at the Museum für Sächsische Volkskunst.

Image Courtesy  Wikipedia Commons

Image Courtesy Wikipedia Commons

The recent winter weather here in North Carolina had us retreating indoors a little sooner than normal and gave me a chance to appreciate the operation of the table top version we have in our home. I found myself wondering about the machine’s efficiency in converting the heat released by the flame into the mechanical energy of the propellers and carousel. So, as a computational engineer is inclined to do, I decided to simulate it using Computational Fluid Dynamics (CFD) in order to satisfy my curiosity.

In thermodynamics, thermal efficiency is the fraction of the energy added by heat (primary energy) that is converted to net work output (secondary energy). In our case, a quick internet search turned up that a burning tea candle releases approximately 50 watts of heat, or 50 Joules of energy per second, which can be used as an input into the CFD model. This work results in an acceleration of the air surrounding the candle flame and, ultimately, the propellers and carousel. The assembly then continues to experience rotational acceleration until such time as the energy input equals the work being done to oppose acceleration by friction. It is important to note that not even an ideal, frictionless engine can convert anywhere near 100% of its input heat into work. This theory is a pillar of thermodynamics known as Carnot’s theory and relates the temperatures at which heat is input and exhausted from the system to the maximum theoretical work, and therefore efficiency of the system as follows:

 
{\displaystyle \eta _{\text{max}}=\eta _{\text{Carnot}}=1-{\frac {T_{\mathrm {C} }}{T_{\mathrm {H} }}}}
 

Thinking about the Christmas pyramid in this context, one can conclude, without the help of simulation, that the Christmas pyramid system will have a very, very low thermal efficiency, since inlet and outlet temperatures will be very nearly identical. But still, we want to know just how low.

The release of heat at the candle tips and wicks is enabled by including the physics of combustion in the simulation. The simple one-step reaction of methane with oxygen forming carbon dioxide and water vapor is used. The fuel (CH4) is input at the upper surface of the candle and entire surface of the wick at the rate needed to achieve 50W of heat per candle, according to the heat content of methane of 1011 BTU/ft. This approximation is justified in this case, as it is not the accurate prediction of flame front physics that we seek, but rather the impact of the heat released on air currents and propeller torques. Fresh air (80% N2, 20% O2) is allowed into the domain via the circumferential surfaces assigned a pressure boundary condition and a mix of air and reaction products is allowed to exit at the top of the domain via an outlet boundary condition. The computational grid, or mesh, consisted of over 6M polyhedral elements and is shown in the figure below.

 
Computational Fluid Dynamics (CFD) Mesh of over 6M Polyhedral Cells Representing the Christmas Pyramid System

Computational Fluid Dynamics (CFD) Mesh of over 6M Polyhedral Cells Representing the Christmas Pyramid System

 

The propeller rotation is enabled by a nifty little CFD technology known as the Dynamic Fluid-Body Interaction method, or DFBI for short. The propeller is embedded in a region of mesh that is allowed to slide via interfaces with a background, static mesh. This rigid body is coupled with the fluid via the fluid boundary coincident with the propeller surfaces and the motion of the body is calculated in response to the fluid forces and moments at those surfaces. The rigid body is assumed to have a moment of inertia of 5E-4 about its rotational axis. We then make the assumption that the system is frictionless and that output work is achieved through an ideal energy storage system where the resisting torque is proportional to the rotational speed through a factor of 6E-4. For comparing power input and output, we recall from Newton’s law of rotational motion that rotational power is calculated as:

 
{\displaystyle P=\tau \omega .\!}
 

A time step of 0.01 s was used and a real-world time of 15 seconds was simulated, though motion reached a relatively steady state within 5 seconds. At each time step, the appropriate transport equations and the rigid body motion equations were iterated upon to convergence using the simple algorithm and an Algebraic Multigrid (AMG) method in the commercial CFD solver from Siemens known as Simcenter (previously STAR-CCM+). The animation below depicts the fluid velocity vectors, propeller motion, propeller skin temperatures and flame temperatures as they evolve with time. Keeping an eye on the power output metric, the viewer will notice that the power output fluctuates between 2.5E-4 and 3.5E-4 Watts. Given that a power input of 300W was used as the basis of the simulation (6 candles times 50 watts/candle), a thermal efficiency of 0.0001% is implied.

We hope you have enjoyed this look at how we use CFD to evaluate the performance of thermal-fluid dynamic systems. If you’d like to receive updates like this in your inbox, be sure to signup for our newsletter.

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