Theoretical automotive math problem

[JCHansen1]

Alright, all you automotive mathematicians... Here's an odd one. I'm writing an article about an engine, and I have a math problem. I believe I figured out the equation that gets me the answer I need, but math was never my strong suit, so I'd like to see if anyone here has something to offer that would prove my proposed equation incorrect.

 

Here's the math problem I'm trying to solve. Roughly how many times over the course of 250,000 miles has the crankshaft turned in an engine getting someone where they need to go. I'm well aware that any solution will be merely an approximation based on inputs that are averages and that the exact number is not easily calculatable as there are too many variables at play. That said, here are my inputs:

 

Let's assume over the course of 250,000 miles, the car's average speed is 38 mph. From a real-world standpoint, I drove my car 38,100 miles last year, logging 1,003 hours to do it, which means my average speed last year was 38 mph. Another assumption I needed to make is the engine's average RPM while running and driving. I decided to use 2,000 RPM as a conservative average. I'm aware that number may be high compared to some with taller geared vehicles, but my daily driver Honda buzzes at 3500 RPM in 5th gear on the expressway at 75 and this car is driven almost entirely on the highway, so in my case, I'm comfortable at a 2k RPM average, and frankly think it may be a little low.

 

250,000 miles / 38 mph = roughly 6,580 hours of run time.

 

6,580 hours X 60 minutes in an hour = 394,800 minutes of run time.

 

394,800 minutes X 2,000 average RPM = 798,600,000 crank revolutions.

 

Give or take 100,000,000 revolutions because of all the potential variables (like my last traffic jam in Nashville, that was probably 100,000,000 revolutions right there) does this math make sense? I think it does, but there's a chance someone much better at math than I is going to chime in with some variable I didn't think of that completely skews my number. I'm reminded of C1 Corvette rev counters. As I understand it, the number in the odometer on the far right advances one number every thousand RPM the engine spins, and to calculate your total actual engine revolutions, you'd need to multiply the number shown in the gauge by 1,000, which would easily put most C1 engines with real world miles on them into the hundreds of millions of rotations.

 

I will say that if my number is close, and I think it is, engines are amazing. The very idea that crank journals (to say nothing of rod journals) can reliably spin hundreds of millions of times against bearings separated by nothing more than a thin film of pressurized oil in the cold of winter, the heat of summer, under hard loads and adverse operating conditions (like me as a teenager) is unbelievable. Just thinking about it makes me want to change the oil in all my cars right now.       

 

What say you?

 

 

Edited February 10, 2022 by JCHansen1 (see edit history)

[SC38dls]

Yes that number is accurate for the assumptions you made. It’s wrong if any of your assumptions are wrong or change based on the vehicle you are driving average rpm’s and speed as you said. This raises the questions regarding the accuracy of your assumptions. So to me not being a mathematician says you can’t make assumptions for an average speed over that long of a distance and get an accurate number but you can get an assumed answer. In my business (a computer service bureau) I always told my employees “if they assume what the client wanted they are wrong. Don’t assume find out what the client wants”. So to me you can’t be right. Sorry. 
dave s 

Edited February 10, 2022 by SC38dls (see edit history)

[JFranklin]

The math as you say is variable for crank revolutions depending on idle and gear selection. That is why they invented an hour meter.

[JACK M]

Sounds about right to me.

 

I pondered the opposite side of this recently when talking about drag cars.

If one of those two thousand HP engine run at lets say 10,000 rpm (probably wrong)

And the race lest say 4 seconds.

Not very many revs during the race.

10000 rpm =167 rps X 4 = about 666 revs per race.

[George Smolinski]

I was doin’ ok with yer cipherin’ til I got ta the gazintas. Ya lost me there.

Edited February 10, 2022 by George Smolinski (see edit history)

[1937hd45]

10 minutes ago, JACK M said:

Sounds about right to me.

 

I pondered the opposite side of this recently when talking about drag cars.

If one of those two thousand HP engine run at lets say 10,000 rpm (probably wrong)

And the race lest say 4 seconds.

Not very many revs during the race.

10000 rpm =167 rps X 4 = about 666 revs per race.

I just looked this up:

 

A Top Fuel car gets to 300MPH in less time than it took you to read this sentence. The crankshaft turns 900 times start to finish in that 1/4 mile run. 

[JACK M]

2 minutes ago, 1937hd45 said:

I just looked this up:

 

A Top Fuel car gets to 300MPH in less time than it took you to read this sentence. The crankshaft turns 900 times start to finish in that 1/4 mile run. 

 

2 minutes ago, 1937hd45 said:

I just looked this up:

 

A Top Fuel car gets to 300MPH in less time than it took you to read this sentence. The crankshaft turns 900 times start to finish in that 1/4 mile run. 

 

And several gallons of nitro?

[JACK M]

2 minutes ago, JACK M said:

 

 

And several gallons of nitro?

 

2 minutes ago, JACK M said:

 

 

And several gallons of nitro?

I wont even try and figure that one, What, A cup of fuel per rev?

Sorry, I didnt mean to hijack. 

Having fun on the forums.....

Edited February 10, 2022 by JACK M (see edit history)

[TTR]

Interesting topic. 

While I don’t have an answer at this moment, I recently came up with similar (but different) engineering math question regarding engine rev’s.

 

Is there a somewhat easy/simple math formula to determine maximum (safe) engine RPM, e.g. “redline”, for stock configuration common, 50+ years old engines for which the manufacturer never provided such information ?

 

Let’s say, for example, some 1950’s Buick (or whatever brand) OHV V8 ?

 

P.S. Apologies for hijacking OP’s thread.

 

Edited February 10, 2022 by TTR (see edit history)

[Bill Boudway]

Safe RPM??

Yes,  128 RPM below the RPM which will have #6 rod part company with the crank and migrate with a big explosion through the crankcase.

[Graham Man]

I was told... the NITRO "high flow" is needed to cool the cylinders so the engine will not melt down.

 

I just timed two Nitro cars getting ready to drag race, with burn out, and backing up they were running just over a minute so I am guessing a few more revolutions.  If I remember correctly the engine is tore down after every pass.

 

I read some incredible number of hours cars a left idling over their lifetime.  Also apparently the big car makers are interested in hours your engine has been running, and now record hours it in your computer, too many hours can void your power train warrantee...

 

I agree with your calculations, but I also agree too many assumptions to get an accurate number. 

[8E45E]

14 minutes ago, Graham Man said:

I read some incredible number of hours cars a left idling over their lifetime.  Also apparently the big car makers are interested in hours your engine has been running, and now record hours it in your computer, too many hours can void your power train warrantee...

One can see for themselves how many hours their vehicle has been idling:  https://forum.studebakerdriversclub.com/forum/your-studebaker-forum/stove-huggers-the-non-studebaker-forum/1881210-new-ford-trucks-with-generators-in-texas

 

Then I kind of wonder now with these on-board optional generators the new Ford trucks offer.

 

Craig

[misterc9]

Manufacturers rated maximum engine horsepower at a specific RPM. Don't exceed that RPM rating for your car and you will be OK. Provided your engine is in good shape.

[Bloo]

14 hours ago, TTR said:

Is there a somewhat easy/simple math formula to determine maximum (safe) engine RPM, e.g. “redline”, for stock configuration common, 50+ years old engines for which the manufacturer never provided such information ?

Yes, sort of, but I can't quote it offhand. It has to do with peak piston speed. There was a rule of thumb. Also, longer stroke engines reach the limit faster.

 

Treating the engine as a black box, there was another rule of thumb, the RPM of the peak horsepower rating for short bursts, and 80% of the RPM of the peak horsepower rating indefinitely. Although this rule of thumb comes from the days when most engines did not have pressure fed bearings, one might tend to be more careful with an engine that lacks pressure feed. Also this assumes the engine is in good shape. If it is tired or unknown, more caution is advised. I believe at least one manufacturer (Chrysler) ran the snot out of some of their pressure fed engines during factory break in, exceeding this rule of thumb by a lot.

 

 

Edited February 11, 2022 by Bloo (see edit history)

[bryankazmer]

peak horsepower rarely occurs at maximum rpm.  Think of the shape of horsepower vs rpm curve  - and each design has it's own curve

[Bloo]

15 hours ago, JCHansen1 said:

Here's the math problem I'm trying to solve. Roughly how many times over the course of 250,000 miles has the crankshaft turned in an engine getting someone where they need to go. I'm well aware that any solution will be merely an approximation based on inputs that are averages and that the exact number is not easily calculatable as there are too many variables at play. That said, here are my inputs:

 

 

This is extremely problematic. You can calculate the revolutions easily based on the tire circumference, gear ratio, and miles. You don't need a time factor for that. A car is not acquiring miles while idling. There are several problems even ignoring the idling. One is that vehicle odometers are notoriously inaccurate. I had one car I ran close to 400k miles. I don't know the exact number without digging in my files, but it was a calculation because the car wore out several odometers in that time. If we assume there could be a +- 10% odometer error, which was common at the time due to a stackup of sloppy factory calibration and variations in tires, and if we assume 380,000 miles on the odometer, the car had something between 342,000 and 418,000 miles on it. Once the mileage gets really high, the odometer becomes meaningless.

 

15 hours ago, JCHansen1 said:

my daily driver Honda buzzes at 3500 RPM in 5th gear on the expressway at 75

 

60mph would be more useful, because the number comes out the same as revs/mile. Revs/mile could also be calculated from wheel rpm x final drive ratio x overdrive ratio.

 

3500 / 75 x 60 = 2800 revolutions per mile. 2800 x 250,000 = 700,000.000 revolutions.

 

But the car would have been running in lower gears part of the time. There is no data. How much will this raise the number? If this were a prewar car I would be willing to ignore low gear, as the periods used are so short. Second gear is oh... 3.42 or so. How long are we in second gear? Not much. I'd say 5%. Of course this wont work for the Honda, the numbers will be significantly higher, but what the hell.

 

700,000,000 x 0.95 = 665,000,000   

 

700,000,000 x 0.05 = 35,000,000  

 

35,000,000 x 3.42 = 119,700,000   

 

665,000,000 + 119,700,000 = 784,700,000 revolutions.

 

And if you want to include idling, you would have to know how long you idled. This varies wildly. How long do you idle? 60% of the time in stop and go city traffic? Maybe 1% on road trips? We have to choose something so maybe 20%. How fast does a Honda idle? I am going to guess 850 RPM.

 

15 hours ago, JCHansen1 said:

which means my average speed last year was 38 mph.

 

RPM = revs/mile at 60 MPH (1 mile =  minute).

 

But you only achieved  an average of 38MPH. So using the average speed (it could introduce horrendous errors, but it is all we have) and your 2800RPM figure:

 

700,000,000 revolutions / 2800 RPM = 250,000 minutes.

 

250,000 minutes / 60 = 4166.7 hours.

 

But that is if it was driven at 60mph, so at 38mph:

 

60 / 38 = 1.58

 

4166.7 x 1.58 = 6583.4 hours

 

6583.4 x 60 = 395,004 minutes

 

Take 20% of that time:

 

395,004 x 0.2 = 79000.8 minutes

 

And at 850RPM idle speed:

 

79000.8 x 850 = 67,150,680 revolutions

 

784,700,000 + 67,150,680 = 851,850,680 revolutions

 

And that is likely low because I assumed 3 speeds and prewar driving habits that definitely do not apply to a modern Honda. No doubt it spent more time in lower gears, adding to the revolutions but not the miles. Also you could take +- 5% of the miles used to calculate, as that is probably fair. Odometers are better than they used to be.

 

As you can see, this is impossible to quantify.

 

15 hours ago, JCHansen1 said:

Just thinking about it makes me want to change the oil in all my cars right now.       

 

What say you?

 

Yes. Me too. And the more wear the engine has, the more it matters.

 

Want an even bigger shock? Calculate how much air an engine uses for one gallon of gas. Gas and air are mixed at a 14.7:1 under most driving conditions. It can be leaner on light loads, and is definitely richer at full throttle, but 14.7 is a good number. But, that is by weight. Gallons are volume. Calculate how many GALLONS of air would be needed to burn one gallon of gas. You probably need to take your altitude into account. Multiply that by how many gallons of gas you burned last year. Then, go clean out your air filter.

 

Edited February 11, 2022 by Bloo (see edit history)

[JCHansen1]

Great contributions Bloo. I knew going in I could certainly be 100,000,000 revolutions or more off. Nevertheless, these are impressive numbers for engines as far as I'm concerned. 

 

Interestingly, I'm well aware of how much air an engine uses, thanks in no small part to the work of Jam Handy. He produced a great 1937 film called "Free Air" about the Chevy carburetor. They did an experiment with a big balloon full of air, 15 cubic feet of it, and metered out three tablespoons of gas. They ran the three tablespoons of gas through the car, and it used all 15 cubic feet of air. The test happens at the 6:50 mark, but the whole video is good. 

 

[George Smolinski]

3 hours ago, JACK M said:

 

I wont even try and figure that one, What, A cup of fuel per rev?

Sorry, I didnt mean to hijack. 

Having fun on the forums.....

The 500 cubic-inch supercharged V8 engine in the Top Fuel dragsters produces an amazing 11,000 horsepower. The combination of air and fuel combined with precise engineering helps produce this amount of power. The average run down the track will use 5-6 gallons of nitro methane fuel. The fuel consumption rate is over 90 gallons per minute for the top fuel dragster engine. The engine would use 16-20 gallons of fuel just to go one mile.  

Edited February 10, 2022 by George Smolinski (see edit history)

[JCHansen1]

That's probably more than 15 cubic feet of air lol!

[wayne sheldon]

I am just happy to see that I am not the only crazy person here that can occasionally obsess over extensive calculations for simple pleasures.

[JCHansen1]

Thanks Wayne! You're in good company!

[37_Roadmaster_C]

If you calculate Pi by hand to the 1000th decimal place, how many pies can you eat while do it?????????????????????????

[Bloo]

About three, but the pieces of pie keep getting smaller and smaller.

 

Edited February 11, 2022 by Bloo (see edit history)

[hook]

The conclusion that I come up with is: Too many people have too much free time to waste. !!!!!

[Guest]

18 hours ago, JACK M said:

 

I wont even try and figure that one, What, A cup of fuel per rev?

Sorry, I didnt mean to hijack. 

Having fun on the forums.....

 Probably even more. If the engine miss fires the ignition on one stroke it would probably hydro lock.

[Bhigdog]

Be an interesting app for some geek to write. Engine RPM sent wireless to your phone's calculator/clock/gps which then calculates..... Rev/min. Rev/mile. Mile/rev. Rev/trip. Rev/gallon. Rev/argument with wife............... Ad Nauseam........Bob

Edited February 11, 2022 by Bhigdog (see edit history)

[Gunsmoke]

Talk about a simple math problem wondering off script (drag racing, nitro, horsepower). JC Hansen1 simply asked "What say you?" Is my math correct. Simple answer is you math and formula are both correct. Your assumptions are also reasonable especially since you explain they are based on your best judgement. In typical "anecdotal" calculations like these (anecdotal meaning based on personal experience and not based on a scientific set of well established norms), "rounding" is used to allow for variables in the likely actual numbers. But I like both your approach and agree with the order of magnitude of your final answer(I would round to 1B revolutions for simplicity). To know that a typical crank would make 1B revolutions in it's lifetime is testament to the quality of materials including lubricants that go into these machines. Now and then you hear of engines that have passed 1M miles, or 4B revolutions!

[Bhigdog]

13 minutes ago, Gunsmoke said:

Talk about a simple math problem wondering off script (drag racing, nitro, horsepower). JC Hansen1 simply asked "What say you?" Is my math correct.

For me one of the more  interesting facets of the forum is when a thread gets a bit frayed. Usually there is quite a bit of interesting info to be gleaned when a thread strays from a rigid interpretation of the subject.

I had no idea a top fuel dragster made 11,000 HP and I spent at least 5 minutes absorbing the graphic details in the "burn out" pix.

I found both bits of this frayed thread far and away more interesting than the original question.

Your results may vary................Bob

[JACK M]

21 hours ago, Graham Man said:

Also apparently the big car makers are interested in hours your engine has been running, and now record hours it in your computer, too many hours can void your power train warrantee...

I have a 21 Ram and I can scroll all kinds of stuff. One of them are engine hours at idle and at certain RPMs, Yes, they can figure out how fast I drive and how long I am idling. Towing and non-towing, How high I have my trailer brakes set, ,,,on and on

Not sure what effect that will have on my warrantee, But so far (less one minor recall) I haven't needed it.

Edited February 11, 2022 by JACK M (see edit history)

[JCHansen1]

Good discussion all around! I don't really mind a thread heading off course. I learned a few things about top fuel dragsters I didn't know and had fun reading the comments. If nothing else, I've convinced myself I'm probably within 100,000,000 revolutions, give or take, of my estimate. 10 percent one way or another is close enough. Thanks for playing everyone. 

[George Smolinski]

2 hours ago, Bhigdog said:

had no idea a top fuel dragster made 11,000 HP and I spent at least 5 minutes absorbing the graphic details in the "burn out" pix

11,000 hp boggles the mind. I had a ‘66 Impala with a 550 hp 468. That’s 20 times more hp than my 66.

[hook]

One thing that really boggles my mind is why do I or anybody else for that matter need to know how many revolutions the crankshaft has turned during the course of it's lifetime. Unless, of course, your locked in a cell with nothing else to do. The pandemic is slowing down, go outside and feel the sun, or snow, whatever.

[JCHansen1]

5 hours ago, hook said:

One thing that really boggles my mind is why do I or anybody else for that matter need to know how many revolutions the crankshaft has turned during the course of it's lifetime. Unless, of course, your locked in a cell with nothing else to do. The pandemic is slowing down, go outside and feel the sun, or snow, whatever.

I can answer that. Since I'm the OP, who better? I drove 38,000 miles last year. About 1000 hours of seat time, throughout the midwest, so I was definitely out and about. That's about my average every year. After all the podcasters I enjoy ran out of words, the books on tape all ended, and I'd been down the Paul Simon freeway (not named for the singer) in southern Illinois for about the seventh time, I couldn't help but wonder about the little engine that gets me there and back again all year long. That car now has 252k on it, and I'd leave Kalamazoo and drive to California tomorrow and think nothing of it. (I'm real fussy about routine maintenance, she's always ready to go). Anyway, after running out of anything else to comtenplate on the expressway after all that time my brain needed something to work on, and now here we are. I think they are astounding numbers, possible only through the genius of precision machining (thanks Henry Leland) and modern oil, which is also almost too good to be true if you think about it. I think it's incredible that modern engines last as long as they do, thats all. 

[hook]

18 hours ago, JCHansen1 said:

I can answer that. Since I'm the OP, who better? I drove 38,000 miles last year. About 1000 hours of seat time, throughout the midwest, so I was definitely out and about. That's about my average every year. After all the podcasters I enjoy ran out of words, the books on tape all ended, and I'd been down the Paul Simon freeway (not named for the singer) in southern Illinois for about the seventh time, I couldn't help but wonder about the little engine that gets me there and back again all year long. That car now has 252k on it, and I'd leave Kalamazoo and drive to California tomorrow and think nothing of it. (I'm real fussy about routine maintenance, she's always ready to go). Anyway, after running out of anything else to comtenplate on the expressway after all that time my brain needed something to work on, and now here we are. I think they are astounding numbers, possible only through the genius of precision machining (thanks Henry Leland) and modern oil, which is also almost too good to be true if you think about it. I think it's incredible that modern engines last as long as they do, thats all. 

The precision machining and modern oil has a lot to do with the modern engines longevity I will agree. However. one of the biggest improvements that our old cars didn't have that our new ones do is filtration. Not necessarily oil filtration but air. Our highways are cleaner than the dirt roads, our air filters keep the little steel grinding particles out and yes our oil keeps the carbon junk in suspension to be filtered out and dumped during oil changes. I think that Henry Leland's Lincolns or Cadillacs for that matter if given the modern oil and especially the air filtration of todays car engines would last just as long. Of course we'll have to do a lot of points, condenser, plugs and tune ups along the way!

[32buick67]

Some more math...

 

If this ad is true that a 1932 Buick can last for 150kmi+, and if the Morlocks and Eloi don't take over the planet, then my car with 51kmi after 90 years is supposed to last another 174 years (sustained avg of 567mi per year).

This could mean:

500+ gallons of oil for annual changes.

7000+ gallons of fuel.

63000+ kg of CO2 emitted(DOE estimates).

69+ million wheel rotations.

 

In the distant future, I sure hope these old cars don't get admonished and banned as illegal to operate.

I started driving a manual transmission car at age 12, and I am delighted my teen kids want to learn how to drive our old manual transmission car.

 

A $64,000 question - how many generations away is the human race until nobody knows, or cares to know, how to drive a manual transmission car?

 

 

 

[Mike "Hubbie" Stearns]

Wouldn’t be twice your calculated number?  A tack reads the rpms of the camshaft. The crankshaft has to turn twice for every one revaluation of the camshaft.  Just my two cents worth. Mike

[Bloo]

A lot of them get turned by the camshaft, but read out in the RPM of the crankshaft.

[Rivguy]

I can imagine that the manufacturer is interested in how much time your car spends idling. The stop/start feature is on most new cars, but it is possible to turn it off, if the owner chooses. This is part of the fuel savings ratings, obviously if you are waiting at a train crossing or backed up intersection, or in stopped in traffic your engine is burning fuel without moving your vehicle. The manufacturer will log all this data and in the future maybe the stop/start feature will always be engaged, just to squeeze out that extra little bit of CAFE ratings. I disabled it on my '17 Flex, I find it annoying, but also I'm old sckool, how long can a starter motor last? My Cadillac Northstar V8 had the starter located under the intake manifold, many modern V8s do also. I went through two starter replacements in 169,000 miles, almost a 1,000 bucks a pop. I sure hope the starters in new cars will last longer.

[f.f.jones]

14 hours ago, 32buick67 said:

A $64,000 question - how many generations away is the human race until nobody knows, or cares to know, how to drive a manual transmission car?

How many generations away is the human race until nobody knows, or cares to know, how to drive a Model T Ford?        (getting closer...)

[Malo48]

16 hours ago, 32buick67 said:

 

A $64,000 question - how many generations away is the human race until nobody knows, or cares to know, how to drive a manual transmission car?

 

 

 

 

A friend sent this to me the other day...