This is a question addressed to all the rotary divinities, self-proclamed or not, of this forum.

I read in numerous places that the e/R (eccentricity over generating radius) is critical for proper operation of a wankel RE.

Can someone explain to me why it is so critical and what range should the ratio be in?

Also, does this critical ratio applies only on Rotary engines or any Wankel-based machinery (such as a compressor for ex.)?

Thanks,

IKN

I may be mistaken, but aren't e and R the determining parameters from where you can calculate the sizes of the epitrochoid chamber as well as the triangular surface of the rotor ?

As I understand, those pieces' curves are difficult to reproduce. And MAZDA probably has some expensive machinery to produce these curves, but with fixed ratios (It's just a thought, I don't know nothing, either ;) ).
Hence the news that MAZDA might produce bigger rotor by only increasing their depth...

(I don't know why, but I'm guessing you're looking at getting more torque ? :D )

Yes. e and R are important variables in creating the trochoid shape of rotary engines. The parametric equations the epitrochoid are as follows, in x-y coordinates -

x=e[cos(3a)]+R[cos(3a)]
y=e[sin(3a)]+R[sin(3a)]

The angle a varies from 0 to 360 degrees.

There's more on the way ;).

To further answer why the eccentricity ratio is critical to performance, let's examine the equation for rounded-flank rotary engine displacement, which is

V=(3)[(3)^(1/2)]w(R^2)(e/R)

w is the rotor width
R is the rotor center-to-tip distance
e/R is the eccentricity ratio

Okay, now that we've settled this, let's look at the equation for power, which is

W[dot]=PVN

P is the brake mean effective pressure
V is the displacement
N is the angular velocity, or in layman's terms, revs.

If you plug in the original equation for V, you will notice that power is directly proportional to the eccentricity ratio (e/R). This means increasing (e/R) increases power, and decreasing it results in lower power (e/R), due to the change in displacement.

This is a question addressed to all the rotary divinities, self-proclamed or not, of this forum.

I read in numerous places that the e/R (eccentricity over generating radius) is critical for proper operation of a wankel RE.

Can someone explain to me why it is so critical and what range should the ratio be in?

Also, does this critical ratio applies only on Rotary engines or any Wankel-based machinery (such as a compressor for ex.)?

Thanks,

IKN

IKN, since you have been such a good help to me, I will help you out :). The eccentricity ratio determines the shape of the housing. It also determines the compression ratio. Eccentricity means that, for example, two circles that do not share the same center point. The rotor and shaft are eccentric to each other for rotary engines, like a spirograph. The eccentricity ratio applies to all Wankel rotary-type machinery, such as an oil pump, compressor, etc. The (e/R) range for flat-flanked (ideal) rotary engines is from 0 to 1/4. Why? Assuming an ideal rotary engine, the rotor housing clearance parameter, d, is given as

d=(R-e)-(e+R/2)=R/2-2e

d is actually the difference between the housing minor radius, R-e, and the distance from the housing to mid flank, e+Rcos60, which is e+R/2 (remember trigonometry ;) ). Setting d=0, the equation goes to (e/R)=1/4, which is the critical eccentricity ratio, or (e/R)[crit]. Using the parametric equations I described above, one can realize that the lower limit for (e/R) is zero, or a circle.

Just when I thought I couldn't feel more stupid someone posts stuff like this and makes me feel even stupider... ;)

Don't feel bad friends. Sorry if the math isn't self-explanatory. We are here to exchange ideas. No one here is stupid :).

A big thank Shelley's Man !
Now, I can go on with the math and picture it better :D

And, POOF, here is another mystery from my friend S.M. Since I don't have the machinery to make a whole new rotary design, I'll be happy with what I have. Thanks for giving me nightmares with the numbers, Shell.

Just kidding around,
Charles

The parametric equations are only critical to the shape, and nothing more. Having an (e/R) of zero means that the two circles are concentric, a circular path; there is no compression, no displacement, no power. Math is fun :D.

Hmmmm. I glad I know smart peeps to explain things like this.

Oh yeah, I forgot. The 3 from the parametric equations comes from the fact that there is a 120 degree separation (the ideal rotor is actually a flat-flanked triangle).

Hey shelleys_man,

In your equasions, the diplacement varies directly with e/R, and with R^2 (simplification of the formula would leave it varying linearly with both R and e).

Therefore, wouldn't the displacement - and consequetially, the power output - then increase when R increases but e remains constant (which would also reduce the ratio e/R)?

Of course, this ignores the real-world power that would be lost to spinning up bigger, heavier rotors......

Don't simplify the equation, because then you would be missing the point of the eccentricity factor. You do make a valid point, because displacement is also proportional to power, which I already explained.

The eccentricity ratio is a ratio between the eccentricity of the shaft with respect to the rotor center-to-tip distance. What do you get when you do simplify the equation for displacement? The result is

V=(3)[(3^(1/2)]weR

What does this equation tell us? It tells us pretty much nothing about what we are trying to find, which is the relationship between displacement and eccentricity ratio.

Finally, before I go to Wal-Mart, here's a link that graphically explains what an epitrochoid really is.

http://mathworld.wolfram.com/Epitrochoid.html

Are you going to Wal-Mart to get a nitrous kit?

Charles

It's pronounced, NOZZ :D.

Of course, this ignores the real-world power that would be lost to spinning up bigger, heavier rotors......

Since I am so fixated on this thread, I wanted to reply to this statement by saying that you are right. My equations are meaningless in the real world. There is a correction for this, through transient analysis, meaning it slowly decreases. In an ideal world, we could measure in steady-state from the equations, but since there is transcience involved, we could actually calculate, or at least accurately guesstimate power, torque, etc. But why go through all of the trouble? Trust me, it's not pretty, which is why we assume steady-state function. This makes plugging and chugging much easier :).

Yes. e and R are important variables in creating the trochoid shape of rotary engines. The parametric equations the epitrochoid are as follows, in x-y coordinates -

x=e[cos(3a)]+R[cos(3a)]
y=e[sin(3a)]+R[sin(3a)]

The angle a varies from 0 to 360 degrees.

There's more on the way ;).

Your formulae must be wrong. It's probably a typo. Just draw a few points to check, they should read something like :
x=e[cos(3a)]+R[cos(a)]
y=e[sin(3a)]+R[sin(a)]


To further answer why the eccentricity ratio is critical to performance, let's examine the equation for rounded-flank rotary engine displacement, which is

V=(3)[(3)^(1/2)]w(R^2)(e/R)

w is the rotor width
R is the rotor center-to-tip distance
e/R is the eccentricity ratio

Okay, now that we've settled this, let's look at the equation for power, which is

W[dot]=PVN

P is the brake mean effective pressure
V is the displacement
N is the angular velocity, or in layman's terms, revs.

If you plug in the original equation for V, you will notice that power is directly proportional to the eccentricity ratio (e/R). This means increasing (e/R) increases power, and decreasing it results in lower power (e/R), due to the change in displacement.


All that you prove here is that power increase with Swept Volume, with intuitively makes sense. It does not help me optimise e/R as I can get any given Swept Volume (thus therotical power) with a wide range of different e/R ratios...

IKN, since you have been such a good help to me, I will help you out . The eccentricity ratio determines the shape of the housing. It also determines the compression ratio. Eccentricity means that, for example, two circles that do not share the same center point. The rotor and shaft are eccentric to each other for rotary engines, like a spirograph. The eccentricity ratio applies to all Wankel rotary-type machinery, such as an oil pump, compressor, etc. The (e/R) range for flat-flanked (ideal) rotary engines is from 0 to 1/4. Why? Assuming an ideal rotary engine, the rotor housing clearance parameter, d, is given as

d=(R-e)-(e+R/2)=R/2-2e

d is actually the difference between the housing minor radius, R-e, and the distance from the housing to mid flank, e+Rcos60, which is e+R/2 (remember trigonometry ). Setting d=0, the equation goes to (e/R)=1/4, which is the critical eccentricity ratio, or (e/R)[crit]. Using the parametric equations I described above, one can realize that the lower limit for (e/R) is zero, or a circle.


Here you're getting closer to what I'm looking for but you only define the absolute limits of e/R.
e/R = 0 gives you a round rotor rotating in a round housing. It's a bit like a piston engine that would have a given bore but no stroke.
e/R = 1/4 corresponds to a tringular rotor with absolutely flat sides.

Note that I have a formula that gives you the curvature radius of the rotor side that will give you complete sealing of the housing at TDC.

Also, any CR can be achieved by machining recesses within the rotor flanks. Although this CR explanation could be further investigated.

Remember the Renesis e/R ratio is 0.142857... which means that getting close to this value should give you a good Wankel IC engine.
But why?
And supposing I'm not interested in the design of a Wankel IC engine, but a Wankel compressor or another Wankel machinery, this 0.143 e/R ratio might not be optimal.

I was actually hoping someone who already know would answer the question fo me, in order to avoid to find it out by myself by a time consuming iterative process.

However, I really appreciate your help, Shelley

Yeah. I fudged on the parametric equations.

I'll try three different e/R ratios for the same Swept Volume to see how it affects CR.

Getting to that actual value for (e/R) is quite hard, well at least with the math I used. Where did you get that number from? It's pretty vague in a sense, that they just put it out like that without explaining why. That value for (e/R) is going to be different for different devices. From the formula, changing R has an effect on the eccentricity ratio. Changing e is also going to have an effect. They are dependent on each other. Also, I forgot to mention since the rotor moves in such a way that it touches the housing, those formulas I mentioned earlier, become

x=e[cos(3a)]+R{cos[a+(2n*pi)/3]}
y=e[sin(3a)]+R[sin(a+2n*pi)]

where n ranges from 0, 1, or 2, which defines the positions of the tips, which are separated by 120 degrees (almost like a triangle).

The eccentricity ratio can be found by dividing the parametric equations for the epitrochoid. How Mazda obtained the 0.143 is a design secret. You can try it out, since you already know the width of the 13B-MSP's rotor, 80 mm.

I forgot to touch on machining the rotor recesses to change the compression ratio. I think of it in terms of a dished and domed piston. Having a flat flank is obviously going to have a higher compression ratio than a recessed flank. The recess increases both minimum and maximum by the same amount, like the dish and domed piston. Also, rotor recesses help improve the shape of the long, narrow combustion pocket forming the minimum swept volume.

I would be interested in knowing why Mazda decided 0.143 as the optimal (e/R). Personally, (e/R) can only be found through experiment.

Here are the results of the variation of CR vs. e/R. Vs remains constant at 654cc (//Renesis). Renesis' Vrp (volume of 1 rotor pocket) estimated at 23cc. Other assumption, I calculated the CR as for a piston engine : CR = (Vs+Vch,min)/Vch,mi (Vch,min = min. comb. chamber volume).

e/R.............CR
0................67 ----- here, rotor and housing = circles
0.01...........16.3
0.05...........13.8
0.1.............11.5
0.12...........10.7
0.142857...10 ------- Renesis
0.16...........9.4
0.18...........8.9
0.2.............8.4
0.25...........7.3 ------- here, Ro (curvature radius of rotor surface) = infinite => rotor with flat flanks

However, with the rotor pockets allowing a wide range of CR anyway, there must be more than just a CR issue in this e/R ratio.

Or maybe is it just the Wankel equivalent to the also critical Bore/Stroke ratio of a piston engine?

I'm as about as clueless as you are about other uses for (e/R). As far as I know, (e/R) determines the shape of the housing, displacement, compression ratio, and performance.

Don't simplify the equation, because then you would be missing the point of the eccentricity factor. You do make a valid point, because displacement is also proportional to power, which I already explained.


But, if the reduced equasion is:

V=(3)[(3^(1/2)]weR,

then doubling both e and R would give a 4X increase in V (and thus power), but would maintain a constant e/R ratio value. Similarly, to say that V varies directly with e/R is misleading, since the ratio would increase with a reduction of R and leaving e constant, but would result in a reduced value for V.

I'm not sure ho much more technical I can get, it's been several years since I had to use any thermodynamics (and all of my propulsion education was focused on turbojets/fans, Ram/SCramjets and rockets). For internal combustion engines, my knowledge in this regard is just the basic theoretical/philosopnical level.

You too? My thermodynamics teacher dejected the Otto cycle, and we just learned power plants, and Brayton-cycle analysis, none of which is related to automotive :(. I think the equation I expressed is very inflexible; I am trying to find a way to make these equations relevant to what IKN wanted to know. My head hurts :(. I hate to sound redundant yet again, but experimentation may better yet help solve this problem. I don't know. My overall knowledge about ICE only goes as far as the basic engineering equations. The developed concepts that come from the equations, I believe, are most important. Thank you for bringing up that point bgreene. I have yet to plug anything into my TI-89 :o.

I think part of it for me was that my propulsion class was through the AE department (since that was my major), and it's been some time since any sort of ICE has been considered the primary power plant for aircraft. Also, all of my fluid dynamics coursework was through AE, and dealt mostly with wind tunnels, aerodynamics, and compressibility (little of which is very useful inside an engine)

We took thermo from the ME dept, but that was just the basic tools and principles (if memory serves, most of the sample stuff actually had to do with steam engine components and before/after states more than any sort of real combustion problems).

On top of that, it's all a bit rusty 'cause I've been doing structural analysis since I graduated in '96, and have had little need to delve into thermo or fluids in that time.

ah well, such is life.....

I am beginning to understand, having read this entire thread....and only God knows why I did that.... why cars cost so much. And all this time I was thinking along the lines of push gas pedal, cars goes. Push gas pedal further, car goes faster, when clearly there is far more to it.

I think I said I read the entire thread. Its true, I did. Understanding it however is another matter altogether. Careful, did I say matter? I certainly don't want to get this thread started on a matter/energy discussion. Forgive me for saying that.

I will say that this thread has been most enlightning. I shall check back often to learn more. Thanks guys!

Well guys, I am still crunching numbers to see if there are any correlations between the eccentricity ratio and other engine parameters. It's hard :(. What I would really like to get my hands on is Kenichi Yamamoto's book, Rotary Engine, to help me further understand how this novelty works :). As an ME student, I have to take two thermodynamics classes. I've taken the first one, which is pretty much power cycles. My second one includes combustion, psychrometrics, you know, stuff normal gearheads never have to worry about :). My friend has my thermo book so I cannot go back and do anything useful until the fall semester starts in August :(. I will do what I can to get to the bottom of this. Thank you everyone for supporting this rather complicated thread :). And thank you, IKN for bringing it up :).

Well guys, I am still crunching numbers to see if there are any correlations between the eccentricity ratio and other engine parameters. It's hard :(. What I would really like to get my hands on is Kenichi Yamamoto's book, Rotary Engine, to help me further understand how this novelty works :). As an ME student, I have to take two thermodynamics classes. I've taken the first one, which is pretty much power cycles. My second one includes combustion, psychrometrics, you know, stuff normal gearheads never have to worry about :). My friend has my thermo book so I cannot go back and do anything useful until the fall semester starts in August :(. I will do what I can to get to the bottom of this. Thank you everyone for supporting this rather complicated thread :). And thank you, IKN for bringing it up :).

Apparently, we ALL want that book...

I wouldn't bring the subject if I didn't need the answer! For the moment being, I'm doing all my modeling work using the Renesis e/R ratio as it is confirmed it mechanically works...

Apparently the R/e ratio is commonly called the K Factor. Does this name ring a bell? I can now re-write my question : how critical is the K factor for a Wankel rotary piston machinery and why (+ recommendations)? Thanks to all.

IKN,

I don't want to inadvertantly insult you by saying this (I don't know how technical your background is), but just about anything that's engineered to a significant extent is likely to have some attribute called "K factor" (at least in USA educated engineers, since "k" and "n" are used in notation as sort of catchall variables in many different kinds of calculations where some sort of coefficient is needed that doesn't match with some otherwise defined value).

I did find some interesting things using google to seach for "k factor" and wankel, including a site with some pics of parts from a 25-liter displacement rotary engine. (searching just on "k factor" returned a list the length of the phone book, with very few sites having anything to do with engines).

From the equasions that shellys_man posted, the total engine compression can, but doesn't neccesarily vary with just the R/e ratio (this would depend on how the ratio is modified). However, the ratio could play directly into compression ratio, or flatness of the rotor faces, both of which could affect engine performance/efficiency.


P.S. bookfinder.com has a line on someone selling a used copy of _Rotary_Engine_. The seller (not me) is asking about $220 for it, though. You can check it out going to www.bookfinder.com and do a title search.

IKN,

I don't want to inadvertantly insult you by saying this (I don't know how technical your background is), but just about anything that's engineered to a significant extent is likely to have some attribute called "K factor" (at least in USA educated engineers, since "k" and "n" are used in notation as sort of catchall variables in many different kinds of calculations where some sort of coefficient is needed that doesn't match with some otherwise defined value).

I did find some interesting things using google to seach for "k factor" and wankel, including a site with some pics of parts from a 25-liter displacement rotary engine. (searching just on "k factor" returned a list the length of the phone book, with very few sites having anything to do with engines).

From the equasions that shellys_man posted, the total engine compression can, but doesn't neccesarily vary with just the R/e ratio (this would depend on how the ratio is modified). However, the ratio could play directly into compression ratio, or flatness of the rotor faces, both of which could affect engine performance/efficiency.


P.S. bookfinder.com has a line on someone selling a used copy of _Rotary_Engine_. The seller (not me) is asking about $220 for it, though. You can check it out going to www.bookfinder.com and do a title search.

Dear bgreene,

Although I indeed have a sufficient technical background, I did not now about the common use of the K Factor terminology. Thanks for clarifying that to me. My searchon Google did not return any satisfactory hit (not in the first 10 pages anyway - shall try again).

Re the effect of this ratio on CR, please refer to one of my previous posts in this thread showing the variation of CR as a function of e/R ratio, for a given swept volume. This ratio indeed also affects the curvature radius of the rotor flanks in order to ensure proper sealing during operation.

I also know about the book at $220- in a US based library. However, I'm first trying to find another,cheaper, source for the book, considering I don't realy know its exact content and technical relevance. Thanks nevertheless.

Here you're getting closer to what I'm looking for but you only define the absolute limits of e/R.
e/R = 0 gives you a round rotor rotating in a round housing. It's a bit like a piston engine that would have a given bore but no stroke.
e/R = 1/4 corresponds to a tringular rotor with absolutely flat sides.

Note that I have a formula that gives you the curvature radius of the rotor side that will give you complete sealing of the housing at TDC.

Also, any CR can be achieved by machining recesses within the rotor flanks. Although this CR explanation could be further investigated.

Remember the Renesis e/R ratio is 0.142857... which means that getting close to this value should give you a good Wankel IC engine.
But why?
And supposing I'm not interested in the design of a Wankel IC engine, but a Wankel compressor or another Wankel machinery, this 0.143 e/R ratio might not be optimal.


Looking over this one quickly, I'm not sure if you've got the right two numbers.

You say that e/R of 0.25 is maximum, however one of my google searches (I'll see if I can find the page again and post a link) made reference to a wankel engine with a R/e value > 25.

Obviously "R" refers to the distance from a rotor tip to the centroid of the "triangle". This would define the circle within which the pure rotation of the rotor could occur.

"e" however, is a geometric property of the eccentric shaft, not the rotor shape, and would determine the circle around which the rotor centroid travels in its planetary motion. For a triangular rotor, the limit for "e" would be the distance from the "flat" to the centroid (minus eccentric shaft radius and whatever minimum wall thickness is required by the strength/durability limits of the rotor). However, "e" could be any value from zero to this limit.

In theory, assuming a flat-sided equilateral triangle rotor with face length of "A",

R= A/(3^.5), and e(max)= A*(3^.5)/6 [this rerults in a R/e ratio = 2.0 at e(max)]

These two values are theoretically independent of each other to some extent, although there may well be many combinations for which the required housing shape would not be producable.

Curvature of the rotor faces, beyond influencing the needed shape of the housing, would also increase the maximum theoretical eccentricity of the engine, since the midpoint of each face would be farther from the centroid of the rotor "triangle".

As with any complex mechanism, theory will only get one so far before real test/operation data is needed. It's likely that a big factor in Mazda's choice of the R/e ratio for the Renesis (at or very close to 7.0) is as much due to this value being one that they have a lot of data points for, since it's very close to or the same as the ratio for previous rotaries that they have made, since extensive study of this variable wold get quite expensive in a hurry with all of the engines they'd have to build.

Just to increase the eccentricity at the current rotor size might require more exotic materials to make the rotors or the eccentric shaft (makes the thing more expensive), could cause durability issues with rotors cracking, might require changes to the depth of the "pocket" in the rotor faces, and would alter the way the gearing on the e-shaft and the rotors would interface, possibly making proper meshing impossible. And those are just the possible issues that I can think of......

Looking over this one quickly, I'm not sure if you've got the right two numbers.

You say that e/R of 0.25 is maximum, however one of my google searches (I'll see if I can find the page again and post a link) made reference to a wankel engine with a R/e value > 25.

R/e = 25 would mean e/R = 0.04, and that's < 0.25 !
e/R of .25 is not an absolute limit. Beyond that point, you get a negative curvature of the rotor flanks. Also, a bit futher beyond that point, the epitrochoid become destorted. Also, well before e/R=0.25, the planetary gear on the rotor is bigger than the rotor itself, wich can lead to design difficulties.



Obviously "R" refers to the distance from a rotor tip to the centroid of the "triangle". This would define the circle within which the pure rotation of the rotor could occur.

"e" however, is a geometric property of the eccentric shaft, not the rotor shape, and would determine the circle around which the rotor centroid travels in its planetary motion. For a triangular rotor, the limit for "e" would be the distance from the "flat" to the centroid (minus eccentric shaft radius and whatever minimum wall thickness is required by the strength/durability limits of the rotor). However, "e" could be any value from zero to this limit.

In theory, assuming a flat-sided equilateral triangle rotor with face length of "A",

R= A/(3^.5), and e(max)= A*(3^.5)/6 [this rerults in a R/e ratio = 2.0 at e(max)]

These two values are theoretically independent of each other to some extent, although there may well be many combinations for which the required housing shape would not be producable.

Curvature of the rotor faces, beyond influencing the needed shape of the housing, would also increase the maximum theoretical eccentricity of the engine, since the midpoint of each face would be farther from the centroid of the rotor "triangle".
In my documents R refers to the epitrochoid generating radius. But indeed it corresponds to the distance you mention above.

e directly and univocally influences the curvature radius of the rotor flanks. You indeed somehow have to respect a given curvature in order to obtain proper 'squish' of the charge after both TDC.


As with any complex mechanism, theory will only get one so far before real test/operation data is needed. It's likely that a big factor in Mazda's choice of the R/e ratio for the Renesis (at or very close to 7.0) is as much due to this value being one that they have a lot of data points for, since it's very close to or the same as the ratio for previous rotaries that they have made, since extensive study of this variable wold get quite expensive in a hurry with all of the engines they'd have to build.

Just to increase the eccentricity at the current rotor size might require more exotic materials to make the rotors or the eccentric shaft (makes the thing more expensive), could cause durability issues with rotors cracking, might require changes to the depth of the "pocket" in the rotor faces, and would alter the way the gearing on the e-shaft and the rotors would interface, possibly making proper meshing impossible. And those are just the possible issues that I can think of......
Yes indeed, so far Mazda seems to have progressed on the rotaries in very careful steps. Geometry of Mazda rotaries have been almost identical for quite a while now.
I'm not intending to design a rotary engine though, but another 'machinery' that would benefit from more freedom in the e/R ratio. However, the e/R criticality, as you mentionned, could have other effects on the rotary design/operation than just the CR. And I'd like to know...