Let me summarize what has been covered and what remains in this efficiency analysis:

After a lot of thought I came up with an approach that I believe is sound to finish this efficiency analysis. In the process I discovered that my heat bookkeeping method would need to be revised. The changes are shown in the above table. The numbers I’ve computed so far weren’t changed, just how they are grouped.

The gross engine power is now grouped as part of the thermodynamic cycle input. The gross engine power is what remains from the thermodynamic cycle input heat after you’ve paid off all your thermodynamic cycle losses. The regenerator is where you save heat from one cycle to the next, so it also belongs as part of the thermodynamic cycle input.

Although I often think of the regenerator losses in comparison with a perfect regenerator, another way to look at it is comparing it with no regenerator. The more efficient your regenerator, the less heat you need to supply overall for the thermodynamic cycle input. A regenerator should never lose heat, an efficient one saves more heat than an inefficient one.

Here are the assumptions and equations:

(1) After elimination of other avenues for heat loss, the only remaining heat is that which goes into the thermodynamic power cycle. If we call the remaining heat Q_available then the thermodynamic cycle efficiency is:

Cycle efficiency = gross engine power / Q_available
Cycle efficiency = 1.6w/22.8w = 7.0%

(2) The heat required in an engine cycle can be computed using the internal energy change for an ideal gas and the PdV work. Air behaves as an ideal gas at the temperatures and pressures we are considering, so this should be a good assumption:

Q_required = U₂ – U₁ + PdV work = mCv(T₂-T₁) ∫₁² PdV

Where: U₂ – U₁ is the internal energy change in joules between two states
m = mass of gas
Cv= heat capacity at constant volume for the gas
T₂, T₁ = temperatures Kelvin of the two states
∫₁² PdV = PdV work between the two states

This computation is complicated by having three different volumes of gas (hot, cold, and regenerator) that are continually changing mass. I used a spreadsheet to compute these values from my simulation results using the following method:

U₂ – U₁ = Cv(Σm₂T – Σm₁T)

The summation includes each of the three masses multiplied by each of the three temperatures. In this case the three temperatures are fixed and the masses in each volume change. Then I sum together all the steps in the simulation where heating takes place (but not cooling) to arrive at a total internal energy increase required of 8.5 j/cycle or 29.8w at 3.5Hz. The internal energy change is the majority of the heating on a low temperature engine such as this one.

The total Q_required can be made more accurate by including the PdV work that occurs during the part of the cycle where heating occurs. Including the PdV work (from the simulator) increases the total heat required to: 9.3 j/cycle or 32.6w at 3.5Hz.

(3)The heat required minus the heat available is the heat provided by the regenerator. This certainly makes sense and is why we use a regenerator. The regenerator saves heat from one cycle to the next so that the engine requires less heat than it would without the regenerator.

Q_required – Q_available = Q_regenerator
32.6w – 22.8w = 9.8w

(4) If the regenerator was 0% efficient, then it provides no power. The thermodynamic cycle efficiency would just be the gross engine power divided by the heat required.

Cycle efficiency = gross engine power / Q_required (0% regeneration)
Cycle efficiency = 1.6w / 32.6w = 4.9% efficiency (without regeneration)

(5) If the regenerator was 100% efficient, then the thermodynamic cycle efficiency would be the Carnot efficiency. This would be a valid assumption if the engine used only reversible processes. Ideal Stirling engines are reversible, you can use them as efficient heat pumps as well as engines. Real Stirling engines are not 100% reversible, they have losses. I think this is a reasonable accurate assumption for the present purpose. Even if a 100% efficient regenerator resulted in 75% of Carnot efficiency, the error introduced in this analysis is 1.7w.

Cycle efficiency = Carnot efficiency (100% regeneration)
Carnot Efficiency = 1- (Tc/Th) Where Th and Tc are absolute temperatures

On my cycle simulator the 1.6w gross engine power corresponds to Th and Tc temperatures of 205.3 C and 58.8C.

Carnot Efficiency = 1 – (58.8 + 273.15)/(205.3 + 273.15) = 30.6%

Thermodynamic cycle input = gross power out / cycle efficiency
= 1.6w/.306 = 5.2w

If the regenerator were 100% efficient, then Q_available would need to be 5.2w. Using assumption (3) then:

Q_required – Q_available = Q_regenerator =
32.6 – 5.2 = 27.4w (100% efficient)

Knowing the heat supplied by a 100% efficient regenerator lets us compute the actual regenerator efficiency:

Q_{regen(actual)} / Q_regen(100%) =
9.8/27.4 = 36% regenerator efficiency

Using the above computations the updated heat loss table is now:

Conclusion

Although I believe the methods and assumptions I used in this part of the analysis are reasonably sound, the regenerator efficiency seems low. I expected something around 80% and have difficulty believing it is below 50%.

Part of the problem may be caused by the possibly inaccurate 22.8w value arrived at by subtraction from the total. For example, a 5 watt increase in estimating the heat lost directly to the environment (a 12.5% error) would result in a 5 watt higher regenerator output and an efficiency of about 54%.

There is also the possibility that the 36% regenerator efficiency is close to correct and I need to design a better regenerator. Because I believe my analysis is not that far off, I think I have to accept this result.

Overall I was surprised by the amount of heat that goes directly from the heater (the light bulb) into the environment. When I say I’m surprised, I don’t mean by the computed result. When I ran the test I could feel considerable heat coming from the insulated cylinder. This large loss was in spite of my efforts to insulate the hot section.

I’ve also tried propane and kerosene (paraffin) burners. As heat sources they are fine for powering this enginge, but the exhaust gases carry away a significant amount of heat and complicate insulating the engine. Another problem is measuring the power (heat). At the low power level this engine uses, I only burn a few grams an hour. I’d need a laboratory grade scale to measure the input power accurately. My energy loss table would also need another entry for heater efficiency.

Rev 1 details:

Section (2) was replaced above. The original is shown below. Because the value for Q_required was changed, it required changing other sections, but only for the value change. The essential change was adding the effect of the PdV work to Q _required.

(2) The heat required in an engine cycle can be computed using the internal energy change for an ideal gas. Air behaves as an ideal gas at the temperatures and pressures we are considering, so this should be a good assumption:

Q_required = U₂ – U₁ = mCv(T₂-T₁)

Where: U₂ – U₁ is the internal energy change in joules between two states
m = mass of gas
Cv= heat capacity at constant volume for the gas
T₂, T₁ = temperatures Kelvin of the two states

This computation is complicated by having three different volumes of gas (hot, cold, and regenerator) that are continually changing mass. I used a spreadsheet to compute these values from my simulation results using the following method:

U₂ – U₁ = Cv(Σm₂T – Σm₁T)

The summation includes each of the three masses multiplied by each of the three temperatures. In this case the three temperatures are fixed and the masses in each volume change. Then I sum together all the steps in the simulation where heating takes place and arrive at a total heat required of 8.5 j/cycle or 29.8w at 3.5Hz.

This is a gamma configuration Stirling engine. I’m powering it with the heat from a 70 watt light bulb so that I can accurately measure the heat input.

The vertical shaft on the left connects to the power piston and the shaft on the right connects to the displacer. The power piston has a 1.25 inch diameter and a 2.5 inch stroke.

You can see the top of the displacer cylinder but the rest of it is covered up by the large metal cylinder packed with insulation. The displacer has a 1.75 inch diameter and a 2.5 inch stroke. The displacer cylinder has a 2 inch diameter and is approximately 12 inches long. The insulating cylinder around the displacer is 8 inches in diameter.

The thermometer measures the temperature near the hot end of the displacer cylinder. It reads 400 Fahrenheit or about 200 Celsius. In the original video the resolution is so poor you can’t read the thermometer. I’ve placed a better resolution photo of the thermometer next to it so you can see the approximate reading was 400 F (although it’s just at room temperature in this photo).

The torque measuring setup shown here is detailed in a previous post. Adjustment screws on the wood torque arm are set to squeeze the rotating shaft to achieve the desired torque value which is read on the scale below. Note that the torque force is transmitted through pins in the wood arm set at a 3 inch horizontal distance that makes the horizontal position of the arm support position on the scale uncritical.

I’m measuring the torque output of the engine with this setup. The scale shows a torque of 68 grams on a 3 inch arm or 68g x 3 inches / 454 (g/pound) =.45 pound-inches of torque. I lift the torque arm briefly to the check the zero reading.

This tachometer displays the engine speed in cycles per second times ten. So 33 means 3.3 hz or 198 rpm. More information on using a bicycle speedometer as a tachometer can be found in the low-cost tachometer post.

The 70w halogen light bulb is hidden up inside the insulating cylinder, but you can see the end of the socket for it here.

The watt meter shows the light is actually using 69 watts.

Here you can see the counter balance weights for the displacer and the crankshaft. The crankshaft drives the lever connecting to the power piston directly. The piston lever pivots freely on the horizontal shaft. A second connecting rod on the crankshaft drives an arm fixed on the horizontal shaft. The horizontal shaft then drives the displacer lever. This mechanism provides the 90 degree phase shift between the displacer and power piston.

The measurements shown are used to compute power and efficiency as follows:

Torque x 2PI = work per revolution
.45 lb-in x 2PI = 2.83 in-lb/revolution

(work/revolution) x revolutions/sec = work/sec
2.83 in-lbs/revolution x 3.3 revolutions/sec = 9.33 in-lb/sec

9.33 in-lbs/sec = .778 ft-lb/sec = 1.05 watts

Efficiency = power out/power in = 1.05 w/ 69 w = 1.52%

The measurements I’ve shown compute to 1.05 watts power output and an efficiency of 1.5%

The promoter claims it must be putting out 100 to 150 watts. It was the usual unloaded engine spinning a flywheel. I thought it looked like a good engine. It was powered by a huge concentrating Fresnel lens. Maybe it was capable of 100 watts, or maybe 5 or 10. The promoter held a stick up against the flywheel and was trying to show how powerfully it rubbed on the stick. It looked like a cave man trying to evaluate an engine.

Why be so crude when it’s really not that hard to test the power output of an engine? The video below demonstrates the setup I used to test my model 3F engine. It uses simple, inexpensive equipment. In this video I’ve gone to some extra effort to measure the efficiency too. For that you need an accurate measure of the heat you are using to run the engine.

I’m not proud of the power output (1.05 watts) or efficiency (1.5%) of my engine. On the other hand, it’s nice to know what I’ve got and what I need to work on to improve it. Those that haven’t tested their engines and just make wild guesses may be surprised by the reality.

In a future post I’ll go into where all that heat goes that doesn’t end up coming out the crankshaft of my engine. A lot of it goes from the heat source into the surrounding environment—even with all the insulation.

More information on the torque measurement and the tachometer is available on this website.

The first photo shows the torque arm assembly mounted on my engine and in use. The torque arm assembly mounts on the engine shaft. Because I don’t want to damage my engine shaft the torque arm slides on its own sleeve that is rigidly attached to the engine shaft with a collar.  As the engine turns the torque arm is held stationary by the vertical column resting on the electronic scale. The torque level is adjusted by increasing the spring force that squeezes the torque arm against the sleeve, increasing friction. By measuring the force on the scale multiplied by the radial distance out on the torque arm I have the torque produced by the shaft. Measuring the RPM using the low-cost tachometer I can compute the power output of the engine. The equations are:

Torque = force x torque arm

Power = Torque x 2π x RPM/60

In my case I want the torque in inch-lbs and I measure the force using a gram scale so:

Torque  (in-lbs) = force (grams) x torque arm (inches) x (1 lb/454grams)

Using this torque value in the power equations yields power in in-lbs/sec.

Making the Measurements

To make the torque measurement I locate the vertical column at the torque arm distance I want. For my measurements I use 3 inches and have a pin installed on the torque arm at this distance. Because the vertical column contacts the pin I don’t need precise lateral alignment on the torque arm. Make sure the torque arm is horizontal when making the measurements. The digital scale is convenient for making these measurements because it has very small displacement unlike a beam balance or spring scale. Raise the torque arm off the vertical column and tare the scale. Lower the torque arm onto the vertical column and read the force on the scale.

To make accurate measurements the torque arm should be balanced on the shaft axis. You can also test this by making torque measurement then letting the torque arm rotate 180 degrees and making another torque measurement. They should be identical. On my torque arm I’ve balanced the asymmetrical weight of the spring with additional washers and have a matching pin on the opposite side of the torque arm.

I experience very little drift in the measurements unless the RPM varies considerably. This only happens at low RPM (under 30 RPM on my engine) when torque levels are high and the flywheel inertia slows down between power pulses.

Torque arm assembly details

The torque arm assembly only requires a few parts. The sleeve is the most complex part and was made using a lathe. It could also be made from metal tubing and two collars. The sleeve is made to have a slip fit on the engine shaft (.25 inch on my engine). The outside diameter of the sleeve has two 5/8 inch diameter shoulders to keep the wood torque arm from sliding in or out on the sleeve as it turns. The diameter where the torque arm rides is ½ inch. The sleeve needs to lock firmly on the engine shaft to transmit the torque. I split the sleeve and used a collar to squeeze the split region against the engine shaft.

The wood torque arm started as one piece of .75×1.5×8 inch wood. I drilled a ½ inch central hole to match the sleeve OD between the shoulders and two holes for the bolts that squeeze the torque arm halves together. I also marked the radial distances out on the torque arm for locating the vertical column when making torque measurements. After that I used a table saw to rip the torque arm in half right through the center of the sleeve hole. If you don’t have a table saw for the ripping operation you could just make the torque arm from two pieces to start with.

I originally had the wood torque arm rub directly on the collar and this worked reasonably well but set up an annoying squeak at low RPM. I’ve since inserted felt which avoids the squeak but requires more squeeze force to get the same friction.

The bolts used to apply clamping force to the torque arm are just threaded rod with nuts and washers sandwiching the lower part of the torque arm. I use barrel nuts on the top for easy finger adjustment. On one side I fix the clamping bolt for an even gap in the torque arm and then make all the torque adjustments on the other side by compressing the spring with the clamping bolt to build force gradually for fine adjustment.

The vertical column used to transmit the force on the torque arm to the scale was made from two pieces of U-channel connected by a threaded rod. The rod is cut to length to make the torque arm horizontal.

Torque and Power Measurement Results

The following two plots show the results of some torque and power measurements on my engine. The data isn’t dead smooth and this could be from several causes. Changing power levels on the engine cause some drift. It’s difficult to keep the temperatures constant on this engine while changing the load. The torque measurements seem reasonably stable except at very low RPM. The RPM measurement gets more difficult at low RPM because it varies throughout a cycle and I have to try to average several RPM readings. One difficulty I’ve experienced is a zero drift on the digital scale. It’s usually quite stable but will sometimes drift a few grams. It’s best to check the zero reading before and after a measurement.  



Higher torque and power measurements

The power levels for my measurements have been less than one watt (8.85 in-lbs/sec = 1 watt) so I don’t have a big power dissipation problem. There is no reason you can’t scale up the design I have used here to measure higher torque and power levels (at higher RPMs too) as long as you pay attention to the power dissipation. I have used an aluminum sleeve for easy machining and good heat conductivity. Although I haven’t tested it, I expect this design should be able to handle at least 2 watts at 100 RPM or higher. Scaling up the design for more contact area between the wood and the sleeve and more exposed surface area on the sleeve to get the heat into the air would be desirable for higher powers. The wood will not conduct heat away very efficiently and will be the weak link. You should be able to smell trouble if you get it too hot. Too much power coming out of my engine for the torque system to handle is a problem I’d love to have. When it happens I’ll let you know how I solve it.

After using my bicycle speedometer-tachometer for a while, I decide to test the Schwinn speedometer that I saw on Amazon for $10. Although it uses the same programming values that I show in the table from my earlier Low-cost Tachometer article, it has some differences that I thought I should point out.

1. The maximum “speed” on the readout is limited to 99.9 (mph or kph). You can accommodate 100 RPM or 1000 RPM or whatever you want to program, but the resolution will remain 1 part in 1000. The earlier Bell Speedometer specifies a limit of 200.0 mph or kph and I assumed a max reading of 199.9.  It turns out to my surprise that the display has another digit and goes well past that. I’ve had mine up to 458.0. Even at the documented 200.0 mph or kph limit you have 1 part in 2000 resolution. So the Bell Speedometer gives a little better resolution if that is an issue.
2. The update time on the Bell Speedometer is one second and the update time on the Schwinn speedometer is two seconds. This has plusses and minuses. If you’re trying to grab an RPM reading quickly as in the Flywheel Spin-down work measurement application, you might want the one second update. For most of my work I prefer the 2 second update because the longer period lets it read lower RPM and have less variation. Using 4 magnets and programming the units so I read 1 kph as 1 RPM the minimum speed on the Bell Speedometer is 15 RPM and on the Schwinn it is 8 RPM. The Schwinn speedometer also has a larger display that is easier to read at a distance although the Bell Speedometer display is quite usable.
3. For either unit don’t plan on using the included magnet for mounting on a bicycle spoke (unless you’re using it for that application). Us a strong magnet and use larger magnet if possible so you can have more clearance between the magnet and the pickup. I use these units to measure RPM in a variety of temporary applications and the larger magnets let me locate the pickup about half an inch from the magnets. Smaller magnets typically require smaller clearances.

Currently the Stirling-cycle engine is producing about one inch-pound of net work per revolution. My Excel-based simulation showed a gross output of 4 to 12 inch-lbs of work per revolution depending on the temperature differential and RPM. The rest of the work is either being lost to friction or my simulation is overly optimistic. I’m using the flywheel spin-down method to measure the engine’s frictional losses and track down the distribution among the various parts of the engine.

An outline of the technique:

1. If possible use a flywheel with plenty of inertia so it will spin down slowly.
2. Compute the total flywheel moment of inertia
3. Set up the configuration to test. I’ll start with only the flywheel and crankshaft, then add various components—piston without compression, displacer, etc.
4. With a tachometer (such as the low-cost tachometer) and stopwatch ready, spin up the flywheel to a suitable RPM, note RPM1, start the stopwatch, let the flywheel spin down to a lower RPM, stop the stopwatch and record RPM2.
5. Make a series of runs over RPM values appropriate for your setup.
6. Use a spreadsheet to compute the energy lost based on the change from RPM1 to RPM2 and the flywheel inertia. Compute the energy lost per unit time (work) versus average RPM.

Flywheel Moment of Inertia

My flywheel is built from 3 components as you can see in the photo. The basic flywheel is an 11.9 inch diameter by 3/16 inch thick aluminum disk. Because I needed more inertia, I added a plywood club to the flywheel. The club supports two ¾ inch bolts with 8 washers each to add mass on an 18 inch diameter.

You can compute the total moment of inertia of your flywheel by adding up the component parts. Here are the equations that are appropriate for a typical flywheel:

Solid disk with constant thickness

I_disk=mR²/2  moment of inertia. For a conventional flywheel the center hub might be modeled as disk.

m= mass, Weight-lb/g

g = acceleration of gravity, 32.2 ft/sec². Note that the pounds you measure on a scale are weight; you need to divide by g to get mass (in the unattractive unit of “slugs”.)

R= radius of disk, ft

Flywheel rim or concentrated mass

I_rim=m(R_ave)²

R_ave= average Radius of a concentrated mass. The concentrated mass could be the two bolts shown in the photo of my flywheel, or it could be the rim on a conventional flywheel. R_ave is measured in ft.

m = mass, Weight-lb/g. This is the total mass of the rim, or in my case, the total mass of the two bolts and washers.

Spokes

I_spoke= m(r_o² – r_i²)/3

r_o = Outer radius of the spoke, ft

r_i = Inner radius of the spoke, ft

m = Total mass of all the spokes, Weight-lbs/g

In this case I can treat the club as a spoke with an inner radius of zero. The total weight of the club takes care of the two sides of the club. This equation assumes the spoke has a uniform mass distribution along its length.

I_total = I_disk + I_rim + I_spoke

Kinetic Energy

The total kinetic energy of a rotating mass is:

KE = I_totalω²/2

ω = rotational speed in radians/sec

Converting to RPM:

KE = I_total(.1047 * RPM)²/2

What we really want is the difference in kinetic energy between two different RPMs:

ΔKE = .0055 * I_total(RPM1² – RPM2²)

If you’ve used the units I’ve suggested the result of ΔKE will be in ft-lbs of work. I usually multiply this by 12 to work in in-lbs for my engine.

The remaining computations:

RPM_ave = (RPM1 + RPM2)/2

Energy lost from flywheel per unit time (ft-lbs/sec):

Work = ΔKE/ (spin-down time)

Set up the test configuration

For a first test configuration I used just the flywheel and crankshaft. This should give you the minimum amount of friction and the plenty of time to measure the RPM and spin-down.

For RPM measurements the bicycle speedometer  gives accurate measurements, especially if you can use four equally-spaced magnets to give you direct RPM readout. It’s important that the magnets be equally spaced or you’ll have varying RPM measurements. The bicycle speedometer I used updates at exactly one second intervals so I use the stopwatch to count out the seconds, starting and stopping the stopwatch when the speedometer reading changes. I don’t use the actual stopwatch reading but the closest second reading—because that’s when the speedometer made the measurement. So 18.13 seconds I record as 18 seconds.

Even with accurately spaced magnets the RPM measurements may vary if the rotating mass isn’t near perfectly balanced (does your flywheel always seek the same rest position)? This is one of the sources of variation in the measurements and why you’ll want to take a series of measurements.

The Results

When you get done you should end up with data you can plot that may look something like this:

This plot is for my flywheel and crankshaft with nothing else attached. So at 60 rpm the total friction from bearings and air friction is about .15 in-lbs/sec.

A series of configurations combined on one chart tells the whole story:

The data is relatively clean for the flywheel configuration but gets fairly noise for the more complete configurations. Running the data through a least-squares curve fit provides equations that are useful for analytical work. The generated curves are shown with the data.

The engine open configuration with no compression includes all the frictional sources except compression and a small amount of air friction on the displacer moving back and forth. At 60 rpm the work lost is about 2.6 in-lbs/sec. The engine has to generate approximately that much power just to turn unloaded at 60 rpm.  

These plots give me a very good idea of the frictional losses in the engine. I still need to account for some missing work. One possibility is compression leakage around the piston and around the displacer shaft seal. I’ll look into those in the future.

I have provided a spreadsheet you can use that will perform the moment of inertia computations from your flywheel measurements. It also provides a table for entering your RPM and timing measurements to generate the energy lost table and plot the results.  

If you don’t have Excel here is a version for Google’s free spreadsheet that you can use for your computations and plotting. You’ll need to save this spreadsheet as your own before you can use it.

Copyright 2008 Doug Conner

I bought the cheapest bicycle speedometer I could find at Kmart. I see Amazon has the same model. Amazon also has a Schwinn model for $10 that should work–but I haven’t tested it.

Tachometer Part 2 –I’ve tested the Schwinn Speedometer and some updates on the Bell Speedometer.

Copyright 2008 Doug Conner

Engine model 3d has actually been running for over a week now, but I finally am taking time to start getting this website up-to-date. You can see more photos in the photo pages.

I’m in the process of performance testing the engine to see how closely its power output agree with my simulation. There are many sinks for losing power including: bearing friction on the rotating shaft, sliding friction on the piston and displacer shaft, compression leakage around the piston and displacer shaft seal, air friction on the air moving around the displacer, and air friction on the spinning flywheel.

My basic and inexpensive instrumentation for this operation includes a stopwatch and a bicycle speedometer used as a tachometer. Using engineering computations and these two instruments I can derive the friction based on how long it takes the unpowered engine to spin down. Here’s how it works.

A flywheel stores energy proportional to the square of its speed (RPM). You can compute the actual energy stored in the flywheel at any RPM. Selecting two RPMs, a high and a low value, you compute the energy difference between the two RPMs. You spin the engine up unpowered to just above the high RPM and start your stopwatch when the RPM drops to the high RPM. The unpowered engine spins down and you stop the time when it reaches the low RPM. You already know the flywheel work difference between the two RPMs (ft-lb, joules, or whatever energy units you want). Dividing by the time you measured on the stop watch gives you the power (ft-lbs/sec, joules/sec, etc.) or rate at which flywheel work is being used up in friction.

A massive flywheel helps you make the measurements accurately and more easily because the flywheel will slow down more gradually. That’s the reason you see the club with massive bolts swinging on the flywheel above. This setup is for under 100 RPM use (don’t use it for high speeds or it could disintegrate with potential for bodily injury). You probably want the spin-down time to be at least 5 seconds and preferably 10 or more to minimize the timing measurement error. You’ll want to make enough spin down measurements so that you can see what the variation is due to timing and RPM measurement accuracy. You can also average the results to get a more accurate results.

The flywheel spin down test can be used with engine components starting with the flywheel and the main bearings it rides on. You add components one at a time, repeating tests to see the power loss due to each component. Using this method you can measure say the piston sliding friction instead of just knowing the friction of the entire engine.

I’ll be adding material for those without the technical background to make the flywheel computations. I’ll also post the method for using a bicycle speedometer as a tachometer. It’s inexpensive and really simple. I use it to measure the RPM on my variable-speed lathe too.