# Stiffening of Tesla turbine rotors

Recently I've begun thinking about experimenting with the construction of a Tesla turbine for steam use. The key issue I can find is that most of the examples out there of such turbines use either optical discs or HDD disks to make the actual rotors. In the case of HDD disks, these work well because they're fairly stiff, but the use limits the size of the turbine to well, the size of the largest common HDD platter (3.5") - that is, unless someone can find me a link to the obsolete 8" platter size. This obviously installs an upper practical boundary on the power level of such a turbine.

One of the key issues against Tesla turbine development historically was the inability to develop both thin and stiffened rotors; while building it thick solves the stiffness issue, it drops overall efficiency.

So, I'm taking the time to actually learn some more about engineering - how is "stiffness" quantified in engineering, and what affects it (other than thickness)? What materials would be good candidates for something such as a Tesla turbine rotor?

• Why do thick disks reduce efficiency? (They may but, if so, why). If you have N disks with N-1 channels the rotor width is N(Wdisk + Wchannel) + Wdisk. If you increase disk widths you increase rotor widths but apart from needing to efficiently lead inlet steam to rotor inlets little else changes. If you want good TT efficiency you want laminar flow which requires small channel widths. This requires low flow per size and most people increase channel widths to increase flow and power and end up with turbulent flow and lower efficiency. – Russell McMahon Jun 25 '15 at 9:48
• Note that the above comes both from calcns and Tesla's own explicit comments on channel widths and efficiency. – Russell McMahon Jun 25 '15 at 9:48
• @RussellMcMahon My primary concern is that increasing efficiency by reduction of channel width would mean for a given amount of steam (and also a given size), I would want to minimize disk width to maintain the same channel area. I suppose "space efficiency" is a better way to describe my concerns. – ecfedele Jun 25 '15 at 10:07
• @RussellMcMahon I'm looking for good efficiency here, so the idea of building multiple smaller turbines at a higher efficiency outweighs the desire to have a single turbine flow a large amount and handle high power. I'm looking at this project through the lens of a prospective solar hobbyist project (parabolic reflector boils water, water turns to steam, etc. etc.) – ecfedele Jun 25 '15 at 10:08
• A Tesla turbine has the obvious advantage of simplicity but, usually is inferior in performance to most alternatives for a given power per size ratio. The TT is greatly affected by channel dimensions compared to other alternatives where this is not a major consideration as boundary laayer interaction is not a major part of their mechanism. For an amateur solar steam the TT sounds a good idea. Then, there's solar Stirling :-). Stirling benefits from high temperatures if you can tolerate them and solar makes high temperatures easy enough. – Russell McMahon Jun 25 '15 at 10:21

I don't know much about the operation of a Telsa Turbine but I can talk in general about what affects the stiffness of an object.

The geometry of the disc and the material of the disc both affect the stiffness of the disc. Smaller radius = stiffer. Thicker disc = stiffer. If you are looking for a trade-off between stiffness and weight then you can employ tricks like ribs or hollow pockets inside the disc. The material property you are looking for is the Young's modulus, $E$. Higher $E$ = stiffer material. You can find a list of Young's Moduli for various materials here:http://www.engineeringtoolbox.com/young-modulus-d_417.html. Modern HDD's are made glass/ceramic composites apparently. If I had to suggest a material and money was no problem, I would say graphene or some other nano-engineered material where you could design the stiffness properties at the molecular level. In terms of affordable materials however, you probably won't be able to find anything better than the HDD platter. Of course you will also need to worry about corrosion resistance, temperature stability, etc. due to the environment in which you are operating.

Stiffness is generally characterized in engineering as the relationship between the force applied to an object and the amount that object deflects: $$\delta = f(F,K)$$ Where $\delta$ is the displacement of the point where the force is applied, $F$ is an applied force and $K$ is the stiffness. In the equation above, $K$ is not necessarily a constant with respect to time or deflection. In fact, it can be a function of many things, including but not limited to the deflection, the rate of deflection, the rate of load application, the temperature of the material, etc. The relationship between force and displacement is often simplified to a linear relationship, where $K$ is assumed to be constant. In this case, K is often called the spring constant. $$F = K\delta$$

If we ignore those more complicated effects, then at its most basic level the stiffness of an object is determined by two things: the material properties and the geometry.

The material property most relevant to stiffness is the Young's modulus, $E$. Higher Young's Modulus = stiffer material. From what I found (https://en.wikipedia.org/wiki/Hard_disk_drive_platter) modern HDD platters are made of a glass/ceramic composite, which I'm guessing would be a proprietary, specially-engineered material for each manufacturer. The Young's modulus for these materials is probably not easily available online. As a comparison however, glass has a Young's modulus of 50-90 GPa. Aluminum is what they used to use for HDD platters, which has a Young's modulus of around 70 GPa. Stainless steel has a Young's modulus of around 180 GPa. If money wasn't an issue then you could make a specially engineered carbon-based structure, like graphene which has a Young's modulus of around 1000 GPa. At the end of the day, you're going to be hard pressed to find a material that is as thin as a HDD platter but stiffer. I don't have a source for this, but in my experience steel can't be machined that thinly (it would get seriously warped due to the heat of the machining process), so you'd need to use other techniques to produce the disc. That's why these things are made with glass and ceramics, which can be assembled with chemical techniques instead of mechanical ones.

The geometry part is where things get really complicated.

Here is a diagram of how I am picturing your situation:

There is a circular disk pinned in the center by some kind of axle. The plate deflects to a new position (dotted line) when a force is applied. If I understand correctly, the stiffness that you are worried about is the "floppy" direction of the floppy disc, i.e. the bending stiffness, as I have drawn it and not the torsional stiffness or compression stiffness.

This is a case of plate bending and here is a Wikipedia article where you can read about how that works. https://en.wikipedia.org/wiki/Bending_of_plates. Yes, the math is very nasty.

To get a better idea of how geometry affects stiffness, I propose a simplified case of cantilever beam bending:

There is a known solution for this case (source: https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory). $$F = \frac{3EI}{L^3}\delta$$

So where does the "thickness", $h$, come in? $I$ is called the area moment of inertia and it is calculated from the cross-section of the beam. In the simple rectangular beam case that I have drawn, $I$ is equal to the following: $$I = \frac{bh^3}{12}$$ Check out my source here: https://en.wikipedia.org/wiki/List_of_second_moments_of_area. That page contains many different cross section shapes. Some are more efficient than others. A cross section that is often used in the construction industry is the I-beam (https://en.wikipedia.org/wiki/I-beam) because it handles bending and shear loads quite well, while also conserving material.

So what can we conclude from the simpler case of cantilever beam bending for our plate bending problem? If you have a solid cross-section for a flat disc, the only ways to improve the stiffness geometrically is to reduce the radius of the disc (analogous to $L$) or increase its thickness (analogous to $h$). If you aren't constrained to solid cross-sections or flat discs then you can use ribs or pockets in your disc to reduce material usage while maintaining stiffness. Again, this is ignoring the more complicated effects that can occur with plate bending.

Maybe somebody else can look into whether or not reducing the radius of the disc will affect the functionality of the turbine. Fluid dynamics isn't my specialty!

Wow. Bit of an essay there. Hope this helps.

• All this ignores the increase in effective stiffness as the rotational speed increases, which can be a large effect - it's quite possible for a thin disk to be 10 times stiffer when it's rotating than when it's not. – alephzero Jan 20 '17 at 20:45

As was previously mentioned, a Tesla turbine is attractive from a simplicity standpoint, but it is not the most efficient solution given current material technology. There are lots of other efficiency considerations on a small project like this that need engineering attention. Don't set your hopes too high; if you get 20% efficiency you are doing very well.

If you operate at high rpm; the blades will be in tension permitting very thin blades. This video of a paper table saw blade qualitatively demonstrates how a very thin member can resist deflection. Composite carbon fiber sheets may be a good option. Laser cut thin steel may be your most cost effective. You may need to have the blades cleaned up on a lathe and/or have the finished rotor professionally balanced.

Steam is a very dangerous form of energy. Always pressure test the system with water and have a properly sized pressure relief installed. If you ever plan on selling systems remember that steam is extensively regulated to protect the public; there are many codes you will need to follow. This is the primary reason you see very few small steam turbines.

I am surprised it had not been done earlier. Constant stress design which is also not far away from stiffest designs are Gauss Bell shaped plates/ shims that thin out at periphery varying according to

$$\frac{t}{t_1} = e^{-(r/r_m)^2}$$

where $t$ is thickness and $r$ radius. However changing air gap width effect for boundary layer has to be a proper trade-off.

Composite design with Kevlar, Carbon cloth /epoxy cut as circular disks omni-directionally placed at shaft center and together cured would produce very high speed rotors. Layer balancing is a tricky affair though.