Automotive Formulas For Engine Cylinder Perssure

[Jason Louison]

Hi! I am a new member to this great Forum! I am so excited to be a part of this wonderful community. Anyway, as the title suggests, I am looking for a way to calculate the pressure in each cylinder of any Four Stroke Engine at any given crank angle of Cylinder One. I have been trying creating a spreadsheet based on a very interesting PDF document I found a couple of weeks ago. One cylinder is fine, but trying to make the formulas for cylinder pressure for each cylinder flexible to change/adapt to the change in degrees per cylinder is getting to be really frustrating. Does anyone know a whole, single function out there for cylinder pressure?

Thanks ~ Jason Louison.

 

[Work Hard Play Hard]

Based on the crank angle alone, no, there is not an equation for pressure. There are many more variables involved beginning with if there is a piston offset or not.

 

[Jason Louison]

Based on the crank angle alone, no, there is not an equation for pressure. There are many more variables involved beginning with if there is a piston offset or not.

Thanks, Removed Unnecessary Data Tables, But I think The Turning Moment Diagram is off.

 

[xxChrisxx]

Assume the forcing profile is the same then just phase each cylinder according to firing order.

I'm on my phone, so can't see the zip properly.

 

[Work Hard Play Hard]

Assume the forcing profile is the same then just phase each cylinder according to firing order.

I'm on my phone, so can't see the zip properly.

My first thoughts also but I don't think that's what he is asking. I think the question is; When cylinder 1 is calculated for pressure at a given crank angle is there an equation that will concurrently give the pressure at the remaining cylinders at that same moment. It's why I brought up offset pistons because then the stroke timing is an additional variable to factor in. Typically offset results in a longer power stroke in terms of time but the velocity of the piston is not linear so it's different at every cylinder at any given single moment. Equations for time are not the same as those for angle. The equation would need to include conversions of variables. From there cylinder head valve timing and air flow and air pressures have to be factored in at every crank angle in relation to the number 1 cylinder in the OP's example. How precise is the equation going to be depends on how far the inclusion of variables is going to go. If there is a single equation for this that covers all engines it's one I've never seen or heard of. I'm not even sure if this is possible to determine from mapping performance through the typical ECU. Not that it wasn't possible. It was a matter of decisions about what needed to be recorded for the family car. Tuner chips even recorded at predetermined crank angles for specific engines and not continuously for the entire 720 degrees of a complete cycle.

 

[xxChrisxx]

How the hell do you view the data? What is the fundamental approach to modelling this? As I can't really make sense of what you've just said. I can't tell if I'm just not getting context.

Fundamentally in every multi cylinder engine; what one piston does (whatever that may be) they all do. If he has a working model for one cylinder, as stated , it's just a case of applying the same thing at a different time.

 

[Jason Louison]

Solved (sort of)! I just had to add some conditional statements that change the value depending on the crank angle for ALL cylinders.

"0°-180°" Argument:
IF(AND($A2<180),0.9×Atmospheric Pressure(psi)

"180°-360°" Argument:
IF(AND(Crank Angle≥180,Crank Angle<360),(Atmospheric Pressure(psi)×((Cylinder Clearance Volume+Cylinder Displacement Volume)÷Total Immediate Cylinder Volume)^Expansion Coefficient)

"360°-540°" Argument:
IF(AND(Crank Angle≥360,Crank Angle<540),(Maximum Pressure(atmospheres)×(Cylinder Clearance Volume÷Total Immediate Cylinder Volume)^Expansion Coefficient),

"540°-720°" Argument:
IF(AND(CrankAngle≥540,Crank Angle<720),1.1×Atmospheric Pressure (psi)

And then a 720°-900°argument, a 900°-1080°, and so on. Combined, these formulas ensures that cylinder pressure is accuratley calculated for all cylinders, no matter what angle they start at or end with!
I would share the spreadsheet, but the file is too large to upload here, unfortunately.

I will upload pictures of the revamped spreadsheet though. Thanks!

 

[Jason Louison]

Some of the graphs I have made so far!

 

[Work Hard Play Hard]

How the hell do you view the data? What is the fundamental approach to modelling this? As I can't really make sense of what you've just said. I can't tell if I'm just not getting context.

Fundamentally in every multi cylinder engine; what one piston does (whatever that may be) they all do. If he has a working model for one cylinder, as stated , it's just a case of applying the same thing at a different time.

I understand but as I said he's asked for an equation to do it all at once. He's also asked for a an existing equation which the equation would need to be universal for every engine. You can't assume the valve timing which effects pressure for every engine is the same. Can't assume combustion time measured in degrees of crank angle is the same for every engine. How many variables in the end are there going to be for a universal equation? If you are aware of an function that does this please let me know.

Getting the pressure at 8 different crank angles and then adding them up is not one equation. Or as the OP put it, one function.

Edit, Jason's last 2 posts weren't here when I started but I stand by what I wrote. Programming a series of arguments for graphing is not the same as a single function for any engine in spreadsheet form.

 

[xxChrisxx]

I think you misinterpreted what he wanted to do. It's pretty clear that he had a single cylinder model he was happy with and just wanted to phase the calculation. This is what he has done by the conditional statements. By universal engine he meant 1,2,4,6,8,12 cylinder etc.

OP your model is effectively turning the PV diagram of the cycle into a torque vs crank angle. Are you going to consider inertia effects and friction?

EDIT: Or have you, I can't tell if inertia is already in there from the graphs. : /

 

[Jason Louison]

I think you misinterpreted what he wanted to do. It's pretty clear that he had a single cylinder model he was happy with and just wanted to phase the calculation. This is what he has done by the conditional statements. By universal engine he meant 1,2,4,6,8,12 cylinder etc.

OP your model is effectively turning the PV diagram of the cycle into a torque vs crank angle. Are you going to consider inertia effects and friction?

EDIT: Or have you, I can't tell if inertia is already in there from the graphs. : /

Yes, there are Inertia Contribution columns in the data tables for each cylinder. The Pressure-Volume diagrams have not been implemented yet, but will be soon :)

 

[xxChrisxx]

Yes, there are Inertia Contribution columns in the data tables for each cylinder.

Is this a run moving 'quasi-statically' then? As that torque profile looks like it's only based on cylinder pressure.

 

[Jason Louison]

Is this a run moving 'quasi-statically' then? As that torque profile looks like it's only based on cylinder pressure.

What do you mean by that?

 

[xxChrisxx]

What do you mean by that?

That is for a four cylinder model. Sum up those four cylinder torque values to get torque @ the crank, and looks at the profile.
Try doing this with the model running with different engine speeds.

 

[Jason Louison]

:D I'm so stoked! This is going so well right now!

 

[Jason Louison]

The question now is how do I distort that p-V diagram given peak torque rpm's and current rpm's...

 

[Jason Louison]

May need to start a new thread. : /

 

[xxChrisxx]

It makes sense to keep it in this thread, as it's a natural progression. Distorting the p-V diagram is where things get 'wooly', and this is where Work Hard Play Hard's comment about not having a single formula is oh so true. There isn't really any way to calculate it by hand.

However, there are some broad assumptions we can make that moves the model closer to reality. For each stage of the process we can introduce assumptions that cause losses. You may want to start by reading the finalpaper.pdf again with the assumptions they make.

eg. Ideal otto assumes isochoric combustion (ie instant prssure raise). - We know this isn't true in reality, the burn begins bTDC and peaks aTDC.

 

[Work Hard Play Hard]

I think you misinterpreted what he wanted to do. It's pretty clear that he had a single cylinder model he was happy with and just wanted to phase the calculation. This is what he has done by the conditional statements. By universal engine he meant 1,2,4,6,8,12 cylinder etc.
: /

Obviously you're right but I'm not sure it's necessarily so clear.

1. I am looking for a way to calculate the pressure in each cylinder of any Four Stroke Engine at any given crank angle of Cylinder One.
2. I have been trying creating a spreadsheet
3. One cylinder is fine, but trying to make the formulas for cylinder pressure for each cylinder flexible to change/adapt to the change in degrees per cylinder is getting to be really frustrating.
4. Does anyone know a whole, single function out there for cylinder pressure?

As I said in my first comment, my first thoughts were the same as yours but #3 suggests otherwise with the other statements.

Have you seen this Jason? It gets referenced and cited often.

http://r.search.yahoo.com/_ylt=AwrBTzvmncRYBIUAczxXNyoA;_ylu=X3oDMTByNXM5bzY5BGNvbG8DYmYxBHBvcwMzBHZ0aWQDBHNlYwNzcg--/RV=2/RE=1489309286/RO=10/RU=https%3a%2f%2fwww.hcs.harvard.edu%2f~jus%2f0303%2fkuo.pdf/RK=0/RS=UrH3DzAaKtGEVFDzfmDAgHUV9CA-

 

[Jason Louison]

Obviously you're right but I'm not sure it's necessarily so clear.

1. I am looking for a way to calculate the pressure in each cylinder of any Four Stroke Engine at any given crank angle of Cylinder One.
2. I have been trying creating a spreadsheet
3. One cylinder is fine, but trying to make the formulas for cylinder pressure for each cylinder flexible to change/adapt to the change in degrees per cylinder is getting to be really frustrating.
4. Does anyone know a whole, single function out there for cylinder pressure?

As I said in my first comment, my first thoughts were the same as yours but #3 suggests otherwise with the other statements.

Have you seen this Jason? It gets referenced and cited often.

http://r.search.yahoo.com/_ylt=AwrBTzvmncRYBIUAczxXNyoA;_ylu=X3oDMTByNXM5bzY5BGNvbG8DYmYxBHBvcwMzBHZ0aWQDBHNlYwNzcg--/RV=2/RE=1489309286/RO=10/RU=https%3a%2f%2fwww.hcs.harvard.edu%2f~jus%2f0303%2fkuo.pdf/RK=0/RS=UrH3DzAaKtGEVFDzfmDAgHUV9CA-

No, I haven't, but it seems like a good resource. Right now I am trying to think of ways to incorporate "losses" in my spreadsheet, so I can get a realistic p-V relationship like this:

 

[Jason Louison]

Here is just a little dummy engine I made: A turbocharged 2.5 Liter Inline Five-Cylinder Engine.

If you guys think anything os off so far let me know! :)

 

[Work Hard Play Hard]

Right now I am trying to think of ways to incorporate "losses" in my spreadsheet, so I can get a realistic p-V relationship like this:

I was wondering if your simulation was for an actual engine or as you say, "a dummy engine?" It might be helpful if you could move on from the dummy engine to an actual engine you can compare your simulation model to. Both the simulation model you referenced here and the one I just added use actual engines to compare their models to. The paper I added deals with contrasts between the model and engine in more depth then your reference and I believe it is a bit more specific and maybe even precise.

Being you're a student I think suggestions at this point are appropriate for identifying, defining and correcting for losses in your model. Again get an actual engine to apply you model to. If you don't your model and coinciding paper will come down to, "I did this because. . . .I think based on. . . ," which probably won't carry as much authority as working with actual deviations between your model and an engine will. If you don't have access to actual data ask for permission to use someone else's. I don't think you'll have much difficulty getting it.

For methodology I would pursue comparison and contrasting. Overlay the actual engine graphs with the simulation and where they match, define why. Where they don't match pursue reasons why they contrast. Document the process for your paper even when you get it wrong because "the wrong," should show up somewhere in your simulation and hopefully you can, or with help from here you can present the effects of your pursuit and explain the error. In the papers you have there are their, "we were wrongs."

I'm not speaking for anyone else and their thoughts or input but I can suggest a couple of places to start.

The compression ratio stated for an engine is usually the "Static" ratio. It's what you use in your model arguments. The compression stroke is presented as BDC to TDC. The intake valve doesn't close until after BDC though. This is the "Dynamic" ratio which is always lower then "Static" ratios. There is no formula for converting cylinder pressure to compression ratio or vice versa. What effect does that have on cylinder pressure in your model?

Maybe Chris can help you with this since I believe this graph is his. What is going on in every other trough in the V8? Why the little skip with this pattern? Jason, how does this torque curve compare to yours?

Chris, have you ever seen a simulation that actually factored in or out, the flywheel? Or further yet, factoring for engine balances? I'm thinking of:

https://en.wikipedia.org/wiki/Engine_balance

 

[Jason Louison]

I was wondering if your simulation was for an actual engine or as you say, "a dummy engine?" It might be helpful if you could move on from the dummy engine to an actual engine you can compare your simulation model to. Both the simulation model you referenced here and the one I just added use actual engines to compare their models to. The paper I added deals with contrasts between the model and engine in more depth then your reference and I believe it is a bit more specific and maybe even precise.

Being you're a student I think suggestions at this point are appropriate for identifying, defining and correcting for losses in your model. Again get an actual engine to apply you model to. If you don't your model and coinciding paper will come down to, "I did this because. . . .I think based on. . . ," which probably won't carry as much authority as working with actual deviations between your model and an engine will. If you don't have access to actual data ask for permission to use someone else's. I don't think you'll have much difficulty getting it.

For methodology I would pursue comparison and contrasting. Overlay the actual engine graphs with the simulation and where they match, define why. Where they don't match pursue reasons why they contrast. Document the process for your paper even when you get it wrong because "the wrong," should show up somewhere in your simulation and hopefully you can, or with help from here you can present the effects of your pursuit and explain the error. In the papers you have there are their, "we were wrongs."

I'm not speaking for anyone else and their thoughts or input but I can suggest a couple of places to start.

The compression ratio stated for an engine is usually the "Static" ratio. It's what you use in your model arguments. The compression stroke is presented as BDC to TDC. The intake valve doesn't close until after BDC though. This is the "Dynamic" ratio which is always lower then "Static" ratios. There is no formula for converting cylinder pressure to compression ratio or vice versa. What effect does that have on cylinder pressure in your model?

Maybe Chris can help you with this since I believe this graph is his. What is going on in every other trough in the V8? Why the little skip with this pattern? Jason, how does this torque curve compare to yours?

Chris, have you ever seen a simulation that actually factored in or out, the flywheel? Or further yet, factoring for engine balances? I'm thinking of:

https://en.wikipedia.org/wiki/Engine_balance

Not quite :( If I set the Cylinder Pressure not to be darn huge maybe I could get some great results like this one. This is a single cylinder engine in my diagram.

So I just need to tweak some parameters and I should be good. As for the "dummy" engine, your absolutely right, so I will now do an engine very close to me: The TOYOTA 2JZ-GTE. (Its even in my profile pic!)

 

[Nidum]

Are you somewhat infinitely confused about your units and values ?

 

[Jason Louison]

Are you somewhat infinitely confused about your units and values?

Yes. if you could read these and tell me why or how they turning moment diagrams are different, and why they are 0 at 90°, that would be a great help.

 

[xxChrisxx]

OP your turning moment is based on cylinder pressure only. The extra 'hump' are caused by the reciprocating inertia of the piston & conrod.

Imagine you hooked the crank up to an electric motor. No firing. You'd get torsional forcing on the crank due to the accelerations of the piston mass.

 

[Work Hard Play Hard]

So I just need to tweak some parameters and I should be good. As for the "dummy" engine, your absolutely right, so I will now do an engine very close to me: The TOYOTA 2JZ-GTE. (Its even in my profile pic!)

An engine you have an affinity for is a good choice. For the graph comparison I was referring more to how "clean" yours is to Chris's actual engine.

Are you somewhat infinitely confused about your units and values ?

Chicken or the egg? Would working on units and values help him more to get to where he wants to go or would helping him get to where he wants to go help with understanding units and values? It's a serious question. Would the visual of units and values at work in a simulation help with understanding the relationships of units and values as the simulation is tuned more to an actual engine or should the relationship between units and values be fully grasped first? What's more efficient? Chris? You sure understood what was being asked before I did.

 

[Jason Louison]

Wow. Turns Out the Formulas I have been using this whole time were wrong. Following this, the spreadsheet has had a series of successful updates! :)

 

[Jason Louison]

Graphs are now more realistic!

 

[xxChrisxx]

More realistic, but not quite right yet. Do that same engine at a few different speeds.

Before you actually do it cosider the following:
What should happen to mean piston speeds?
What would this do to piston acceleration, hence forcing?
What would you expect to see to the torque cyclic graph?

 

[RogueOne]

Is your piston acceleration and velocity relative to crank angle shown as being sinusoidal on that graph?
Piston Displacement = √((con rod length)*2 + (crank radius)*2 - 2lrcos(con rod angle))

 

[Jason Louison]

Hi guys, I have been doing updates to the spreadsheet all day, so I remember very little about the specs of that engine. However, if its piston position, velocity, and acceleration you want to look at, here are a few graphs all for the same engine. all data logged on the graphs are taken at 3000 Revolutions Per Minute.

Engine: TOYOTA 2JZ-GTE
Specifications
Engine Type: Inline Six Cylinder Engine
Valve-train: DOHC w/wo VVT-i
Aspiration: Sequential Twin Turbochargers
Bore: 86 mm
Stroke: 86 mm
Connecting Rod Length: 135mm-145mm (145mm used in graphs)
Compression Ratio: 8.5:1

*Note: If the graphs do not display all of the cylinders in the engine, it is probably because they are at the same position/velocity/acceleration as the one/all of the cylinders.

Ex: Inline Six
720÷6=120
120*3=360, Cylinder Four is at the same position as Cylinder One, Because it's crank angle is spaced one revolution after cylinder one. I don't know how to solve this

 

[xxChrisxx]

Why do you need to solve it? Two pistons will have the same inertia component. They will be be different when you include the cylinder pressure information as they will be on different strokes.

Cylinder 4 wouldn't be the same as cylinder 1 though in an I6. Look at an animation of an I6, the cylinders should be phased according to firing order. 153624.
Cyl1=Cyl6
Cyl2=Cyl5
Cyl3=Cyl4

 

[Jason Louison]

does anyone know what unit of measure would the bottom equation be in?

((P–Patm)*A*r)÷(sin(a+p))/cos(p))

where P = Cylinder Pressure (psi)
Patm = Atmospheric Pressure (psi)
A = Piston Crown Area (mm^2)
r = Crankshaft Radius (mm)
a = Crankshaft Angle (deg)
p = Piston-Rod Angle (deg)

 

[xxChrisxx]

What are you using that equation to plot? What units do you think it should be in?

 

[Jason Louison]

does anyone know what unit of measure would the bottom equation be in?

((P–Patm)*A*r)÷(sin(a+p))/cos(p))

where P = Cylinder Pressure (psi)
Patm = Atmospheric Pressure (psi)
A = Piston Crown Area (mm^2)
r = Crankshaft Radius (mm)
a = Crankshaft Angle (deg)
p = Piston-Rod Angle (deg)

It plots the pressure contribution of the cylinder.

 

[xxChrisxx]

Hmm, I'm not comprehending something here.

Are you asking what the unit of the equation you wrote are in?

 

[Jason Louison]

Hmm, I'm not comprehending something here.

Are you asking what the unit of the equation you wrote are in?

Yes. I'm confused because the pressure volume diagram plots cylinder pressure as Newtons/meter/meter, Which I think is megaPascals, and now somehow the pressure contribution, Inertia contribution, and Turning Moment are all in Newton Meters.

 

[xxChrisxx]

I think ultimately you need to be much more careful with your units as Nidum said. Note pressure as N/m^2 rather than N/m/m (which is confusing).
The answer is all in the equation. Ignore units for now and conceptually break down the equation into steps.

What is happening at each step:
1. (P-Patm)
2. (P-Patm)*A
3. ((P-Patm)*A)*r

What do the cos and sin terms do to this?

Or try doing this from first principles to gain a conceptual understanding. We are plotting torque yes?
T=F*d
So how to F and d change as we go through the engine cycle? Start with d first.

 

[Jason Louison]

I think ultimately you need to be much more careful with your units as Nidum said. Note pressure as N/m^2 rather than N/m/m (which is confusing).
The answer is all in the equation. Ignore units for now and conceptually break down the equation into steps.

What is happening at each step:
1. (P-Patm)
2. (P-Patm)*A
3. ((P-Patm)*A)*r

What do the cos and sin terms do to this?

Or try doing this from first principles to gain a conceptual understanding. We are plotting torque yes?
T=F*d
So how to F and d change as we go through the engine cycle? Start with d first.

So far, this is what iv'e got:
(P-Patm)
Where
P= Cylinder Pressure (atm)
P= Atmospheric Pressure (atm)

atm–atm=atm

atm*A

Where A= Piston Cross Sectional Area (mm^2)

atm*mm^2= atm*mm^2

atm*mm^2*r

Where r= Crankshaft Radius (mm)

atm*mm^2*mm= atm*mm^3

Now this is were things start to get looney

atm*mm^3*(sin(ac+acr)/cosine(acr))

Where

ac= crank angle
acr = angle between bore centerline and connecting rod

 

[xxChrisxx]

You need to abstract away from the units. They are actually irrelevant to what's going on.

1. Is net (gauge) cylinder pressure. Can be Psi, Bar, atm, MPa, whatever.
2. Pressure * area = ?
3. ^_^ * distance = ?

Draw the crank throw and conrod vertical. If you apply pressure here, what rotating torque do you have?

 

[Dr.D]

The proper phasing depends upon the geometry of the engine, specifically (1) the relative crank angles and (2) the relative cylinder inclination angles (in-line, V, opposed, etc). All of this goes into the kinematics that relate one reference crank angle to the effective crank angle for each other cylinder.

Depending on the purpose for the analysis, it may, or may not, be appropriate to simple add the torques from all cylinders.
(1) If the crank is to be considered rigid, then all of the torques are additive.
(2) If the crank is to be considered as flexible, as is necessary for a torsional vibration analysis of the machine, then each cylinder torque should be considered as a separate excitation, with appropriate phasing.

 

[Jason Louison]

You need to abstract away from the units. They are actually irrelevant to what's going on.

1. Is net (gauge) cylinder pressure. Can be Psi, Bar, atm, MPa, whatever.
2. Pressure * area = ?
3. ^_^ * distance = ?

Draw the crank throw and conrod vertical. If you apply pressure here, what rotating torque do you have?

at 360°, where pressure=atm, area=mm^2, and distance= mm, I have 85864 atm's*mm^3

 

[xxChrisxx]

You're missing the point of what I'm asking and seemingly blindly applying an equation you have found. To understand properly you need to know how to derive the equation for torque.

Draw(or google) the picture of a crank slider with the piston at TDC.

 

[Jason Louison]

Ignore the values and the figure on the right please.

 

[Nidum]

Recommended reading

 

[xxChrisxx]

So in that left hand image, imagine we apply maximum cylinder pressure (as happens at TDC on an ideal otto cycle). At that instant, how much torque does it make?

 

[Jason Louison]

So in that left hand image, imagine we apply maximum cylinder pressure (as happens at TDC on an ideal otto cycle). At that instant, how much torque does it make?

I can't specify...

 

[xxChrisxx]

You can, its one of the cases where you don't need to see any units or values.

If torque = force * distance?

Force acts vertically downwards, what is th length of the moment arm?

 

[Jason Louison]

You can, its one of the cases where you don't need to see any units or values.

If torque = force * distance?

Force acts vertically downwards, what is th length of the moment arm?

One meter.