Exhaust Gas Pressure/Blowdown Calculations

[Jason Louison]

Hi! I am a bit confuzzled by these equations given by a highly referenced and cited paper I have been using to create a spreadsheet I have been working on. The equations are:

PV=mRT

Where P is the cylinder pressure, m is the mass of gasses in the cylinder, R universal gas constant of the gas, and T is the Cylinder Temperature. The second equation is

T/(P^((y-1)/y))=C

Where T is cylinder temperature, P is cylinder pressure, y is the ratio of specific heats for the exiting gas, and C is a constant given by the Temperature, Pressure, and ratio of specific heats.

I know that if I have the Initial Temperature, Initial Pressure, and Ratio of specific heats, I can compute the constant, and then use the constant, initial, temperature, and the ratio of specific heats to find cylinder pressure, but I'm not getting the results I was expecting. I tried playing around with some other variables, but it still did not yield any logical or sensical results. Here is the PDF of the document.

https://www.hcs.harvard.edu/~jus/0303/kuo.pdf

 

[scottdave]

Temperature must be expressed in absolute (either Kelvin or Rankine). That may be your issue. I am just speculating, not knowing your data values.

To convert Fahrenheit to Rankine, add 459.67 and add 273.15 to Celsius will give you Kelvin.

 

[Jason Louison]

Temperature must be expressed in absolute (either Kelvin or Rankine). That may be your issue. I am just speculating, not knowing your data values.

To convert Fahrenheit to Rankine, add 459.67 and add 273.15 to Celsius will give you Kelvin.

This is what's confusing me. What do they mean by pressure at the throat? How can cylinder pressure or cylinder temperature be a variable? Isn't Cylinder Temperature Dependent on Cylinder Pressure? Isn't that (Pressure) what we are trying to look for? Even if we rearrange the equation, we still have an unknown mass flow rate, which, coincidentally, is dependent on cylinder pressure and temperature!...?

 

[boneh3ad]

First, make sure you use absolute temperatures (Kelvin or Rankine) and pressures (referenced to zero, not to ambient). These equations are thermodynamic in nature, and require absolute measures of those quantities.

Second, I don't know why that image you just posted lists the pressure at the throat since it doesn't appear in the equation. That said, it's a pretty easy quantity to calculate given the upstream conditions and the contraction ratio from the reservoir to the throat.

Third, temperature is related to the pressure (and density, for that matter) via the ideal gas law, ##p=\rho \bar{R} T## or ##p = m R T## according to your paper you are citing. Given that your cylinder is discharging air, the pressure and temperature inside will therefore be a function of time, and therefore variable quantities. How they relate to one another will also depend on your cylinder's interaction with its environment. The second equation you mentioned, ##T/p^{(\gamma-1)/\gamma} = C##, is based on assuming the flow is isentropic, meaning it is both adiabatic and has no dissipation.

Last, the mass flow rate does depend on the pressure and temperature (and throat size), and yet the pressure is also dependent on the mass currently inside the cylinder, and therefore the mass flow rate. This is an example of a real-world situation where, to solve the problem, you need to employ differential equations.

 

[D.B.SriHridai]

Don't worry, in general practical values differs from the theoretical calculations. Because it involves some loses. And more over we assume the specific heat of the gas inside cylinder remains constant throughout the process. But practically specific heat of the gas changes with respect to the change in the temperature in the cylinder. Make sure that all the parameters in the equation are taken in unique system i.e. either in F.P.S. or S.I.

 

[Jason Louison]

First, make sure you use absolute temperatures (Kelvin or Rankine) and pressures (referenced to zero, not to ambient). These equations are thermodynamic in nature, and require absolute measures of those quantities.

Second, I don't know why that image you just posted lists the pressure at the throat since it doesn't appear in the equation. That said, it's a pretty easy quantity to calculate given the upstream conditions and the contraction ratio from the reservoir to the throat.

Third, temperature is related to the pressure (and density, for that matter) via the ideal gas law, ##p=\rho \bar{R} T## or ##p = m R T## according to your paper you are citing. Given that your cylinder is discharging air, the pressure and temperature inside will therefore be a function of time, and therefore variable quantities. How they relate to one another will also depend on your cylinder's interaction with its environment. The second equation you mentioned, ##T/p^{(\gamma-1)/\gamma} = C##, is based on assuming the flow is isentropic, meaning it is both adiabatic and has no dissipation.

Last, the mass flow rate does depend on the pressure and temperature (and throat size), and yet the pressure is also dependent on the mass currently inside the cylinder, and therefore the mass flow rate. This is an example of a real-world situation where, to solve the problem, you need to employ differential equations.

Can you guide me as to what exactly I need to differentiate? The PDF mentioned that too, but I just don't know where to start.

 

[boneh3ad]

Can you guide me as to what exactly I need to differentiate? The PDF mentioned that too, but I just don't know where to start.

Well, lay out what you know. You know an expression for ##\frac{dm}{dt}## in the bottle and you know an expression relating ##p##,to ##\rho## and ##T## in the bottle and a way to relate density to the mass in the bottle, right? So therefore you should be able to find something that looks like a differential equation for ##p## as a function of time.

 

[Jason Louison]

Like this one?!

 

[Jason Louison]

Here are the data in my spreadsheet so far. I am so close!

 

[Jason Louison]

Well, lay out what you know. You know an expression for ##\frac{dm}{dt}## in the bottle and you know an expression relating ##p##,to ##\rho## and ##T## in the bottle and a way to relate density to the mass in the bottle, right? So therefore you should be able to find something that looks like a differential equation for ##p## as a function of time.

I have uploaded a few photos, but I'm still not quite catching on.

 

[jack action]

View attachment 203291

This is what's confusing me. What do they mean by pressure at the throat?

In the document, it says (after equation 6):

The pressure at the throat is equal to that of the gases in the exhaust system.

How can cylinder pressure or cylinder temperature be a variable? Isn't Cylinder Temperature Dependent on Cylinder Pressure? Isn't that (Pressure) what we are trying to look for?

In this case, we are looking for the mass flow rate going out of the cylinder. All pressures and temperatures should be already known at this point.

Even if we rearrange the equation, we still have an unknown mass flow rate, which, coincidentally, is dependent on cylinder pressure and temperature!...?

Once you have found the mass flow rate for a predetermined amount of time (or crankshaft revolution), you can find out how much gas got out of the cylinder (mass-wise). So, you also know how much is left inside the cylinder. For the next iteration, you will be able to estimate a new ##PV = mRT##. You rinse and repeat until you go through the entire exhaust process, step-by-step, one increment of crankshaft revolution at a time.

 

[Jason Louison]

In the document, it says (after equation 6):In this case, we are looking for the mass flow rate going out of the cylinder. All pressures and temperatures should be already known at this point.

Once you have found the mass flow rate for a predetermined amount of time (or crankshaft revolution), you can find out how much gas got out of the cylinder (mass-wise). So, you also know how much is left inside the cylinder. For the next iteration, you will be able to estimate a new ##PV = mRT##. You rinse and repeat until you go through the entire exhaust process, step-by-step, one increment of crankshaft revolution at a time.

My problem is that my Temperature Equation is dependent on cylinder pressure, and I get an error because I don't have the values for exhaust pressure yet. And is the "throat" the throttle body? If so, then it's saying that the pressure at the throat is equal to the pressure of the gasses in the exhaust system, but that would just make exhaust pressure constant, as the equation for intake pressure that I use is a constant... What I'm ultimately looking for is Exhaust Blowdown pressure. This is usually where the curve after expansion and leading to induction is on the P-V diagram.

 

[jack action]

My problem is that my Temperature Equation is dependent on
cylinder pressure, and I get an error because I don't have the values for exhaust pressure yet.

In your OP, you have 2 equations with 2 unknowns, namely pressure & temperature. These equations represent the conditions inside the cylinder. You can then find these values easily.

If you exhaust directly to the atmosphere, you can assume the exhaust pressure (the one in the exhaust pipe, i.e. at the back of the exhaust valve) is the atmospheric pressure.

And is the "throat" the throttle body?

No. It is the "throat" formed by the exhaust valve opening, i.e. the smallest area where the fluid flows.

This opening will determine how much gases exit the cylinder, giving you a new value for ##m## inside your cylinder for ##PV = mRT##.

 

[Jason Louison]

In your OP, you have 2 equations with 2 unknowns, namely pressure & temperature. These equations represent the conditions inside the cylinder. You can then find these values easily.

If you exhaust directly to the atmosphere, you can assume the exhaust pressure (the one in the exhaust pipe, i.e. at the back of the exhaust valve) is the atmospheric pressure.

No. It is the "throat" formed by the exhaust valve opening, i.e. the smallest area where the fluid flows.

This opening will determine how much gases exit the cylinder, giving you a new value for ##m## inside your cylinder for ##PV = mRT##.

I found this in a PDF I downloaded awhile ago:

Can you tell me exactly what I'm looking at here? Can I use an arbitrary value for P_T, like 1 atmosphere? Also, is P₀ the variating cylinder pressure, as well as T₀? How would I differentiate this?!

 

[jack action]

Can you tell me exactly what I'm looking at here?

The exact same equations as (5) and (6) in your previous reference (kuo.pdf). I'm not sure what you are not understanding.

Can I use an arbitrary value for PT, like 1 atmosphere?

It's not arbitrary, if you exhaust into the atmosphere, then it is 1 atm.

Also, is P0 the variating cylinder pressure, as well as T0?!

Yes.

How would I differentiate this?!

This is what you need to do:

1. Define the angular position ##\theta_1## of the crankshaft;
2. Find ##P_0## & ##T_0## inside the cylinder with the 2 equations from your OP;
3. Find ##\dot{m}## at the exhaust valve;
4. Define a crankshaft displacement ##\Delta\theta##;
5. Find the mass ##m## of the exhaust gases that escaped from the cylinder during crankshaft displacement ##\Delta\theta##;
6. Set the new angular position ##\theta_2## of the crankshaft equals to ##\theta_1 + \Delta\theta##;
7. Repeat calculations with new crankshaft position and gas mass inside cylinder.

 

[Jason Louison]

The exact same equations as (5) and (6) in your previous reference (kuo.pdf). I'm not sure what you are not understanding.

It's not arbitrary, if you exhaust into the atmosphere, then it is 1 atm.

Yes.

This is what you need to do:

1. Define the angular position ##\theta_1## of the crankshaft;
2. Find ##P_0## & ##T_0## inside the cylinder with the 2 equations from your OP;
3. Find ##\dot{m}## at the exhaust valve;
4. Define a crankshaft displacement ##\Delta\theta##;
5. Find the mass ##m## of the exhaust gases that escaped from the cylinder during crankshaft displacement ##\Delta\theta##;
6. Set the new angular position ##\theta_2## of the crankshaft equals to ##\theta_1 + \Delta\theta##;
7. Repeat calculations with new crankshaft position and gas mass inside cylinder.

I'll give it a try!

 

[Jason Louison]

The exact same equations as (5) and (6) in your previous reference (kuo.pdf). I'm not sure what you are not understanding.

It's not arbitrary, if you exhaust into the atmosphere, then it is 1 atm.

Yes.

This is what you need to do:

1. Define the angular position ##\theta_1## of the crankshaft;
2. Find ##P_0## & ##T_0## inside the cylinder with the 2 equations from your OP;
3. Find ##\dot{m}## at the exhaust valve;
4. Define a crankshaft displacement ##\Delta\theta##;
5. Find the mass ##m## of the exhaust gases that escaped from the cylinder during crankshaft displacement ##\Delta\theta##;
6. Set the new angular position ##\theta_2## of the crankshaft equals to ##\theta_1 + \Delta\theta##;
7. Repeat calculations with new crankshaft position and gas mass inside cylinder.

Would I be okay using these values for P₀ and T₀? I had already had them calculated before hand.

 

[jack action]

It depends what «at end of expansion» refers to. Usually, the exhaust valve opens before bottom dead center, so the exhaust process begins before the expansion process is completed.

 

[boneh3ad]

Man, all this iteration... you can solve this analytically. I once gave essentially this problem as an assignment in a compressible flow course.

 

[jack action]

Man, all this iteration... you can solve this analytically. I once gave essentially this problem as an assignment in a compressible flow course.

But won't the engine cylinder volume and valve opening constantly changing over time makes this more difficult to solve analytically?

 

[boneh3ad]

But won't the engine cylinder volume and valve opening constantly changing over time makes this more difficult to solve analytically?

It's possible I missed pieces of the problem in this thread as a result of skimming during this end of semester period. However, I'd the open and close time of the valve is neglected, then it should still be fairly easy to solve over the times the valve is open.

 

[boneh3ad]

Alright, for "fun" I went through and derived a differential equation for pressure assuming that the pressure, temperature, cylinder volume, and throat area could all be varying in time assuming an isentropic system. It's... not the nicest equation I've ever seen. Certainly solvable on a computer though. It's a first-order, nonlinear ODE (the partials drop out due to the above assumptions) and basically treats ##V##, ##dV/dt## and ##A^*## as forcing functions. In the case of an engine, ##V## and ##A^*## would also be related to one another.

So it certainly isn't solvable analytically as I originally claimed now that I've re-read through the thread and understand the problem that was actually being solved. If you get rid of the variable ##V## and ##A^*## then it is a solvable equation.

 

[Jason Louison]

Alright, for "fun" I went through and derived a differential equation for pressure assuming that the pressure, temperature, cylinder volume, and throat area could all be varying in time assuming an isentropic system. It's... not the nicest equation I've ever seen. Certainly solvable on a computer though. It's a first-order, nonlinear ODE (the partials drop out due to the above assumptions) and basically treats ##V##, ##dV/dt## and ##A^*## as forcing functions. In the case of an engine, ##V## and ##A^*## would also be related to one another.

So it certainly isn't solvable analytically as I originally claimed now that I've re-read through the thread and understand the problem that was actually being solved. If you get rid of the variable ##V## and ##A^*## then it is a solvable equation.

Can I see a graph of the function?

 

[Jason Louison]

The exact same equations as (5) and (6) in your previous reference (kuo.pdf). I'm not sure what you are not understanding.

It's not arbitrary, if you exhaust into the atmosphere, then it is 1 atm.

Yes.

This is what you need to do:

1. Define the angular position ##\theta_1## of the crankshaft;
2. Find ##P_0## & ##T_0## inside the cylinder with the 2 equations from your OP;
3. Find ##\dot{m}## at the exhaust valve;
4. Define a crankshaft displacement ##\Delta\theta##;
5. Find the mass ##m## of the exhaust gases that escaped from the cylinder during crankshaft displacement ##\Delta\theta##;
6. Set the new angular position ##\theta_2## of the crankshaft equals to ##\theta_1 + \Delta\theta##;
7. Repeat calculations with new crankshaft position and gas mass inside cylinder.

SO CLOSE! But I don't think that the function is valid, because it depends on volume starting at the very beginning of the cycle, and not the instantaneous volume.

UUGGHHH!

 

[boneh3ad]

Can I see a graph of the function?

I haven't plotted it because I don't have enough information at this point about the cylinder stroke or valve opening characteristics. The other thing that hasn't really been addressed is air and fuel injection.

 

[RogueOne]

But won't the engine cylinder volume and valve opening constantly changing over time makes this more difficult to solve analytically?

Yes. Also side load on the piston will change the way that the piston surface interacts with the cylinder wall surface. Thermal expansion throughout the gradient of the block, head, and piston will change it. Crankcase pressure throughout this process will be a variable. The structural behavior of the piston as pressure is applied will be a variable. Oil film on the cylinder walls will be a variable.

Its a very ambitious phenomena to actually calculate, no doubt.

 

[Jason Louison]

In your OP, you have 2 equations with 2 unknowns, namely pressure & temperature. These equations represent the conditions inside the cylinder. You can then find these values easily.

If you exhaust directly to the atmosphere, you can assume the exhaust pressure (the one in the exhaust pipe, i.e. at the back of the exhaust valve) is the atmospheric pressure.

No. It is the "throat" formed by the exhaust valve opening, i.e. the smallest area where the fluid flows.

This opening will determine how much gases exit the cylinder, giving you a new value for ##m## inside your cylinder for ##PV = mRT##.

Got a new book! Hopefully this will clear up some of the contreversy, but I will keep this thread going and post about newly acquired information!

 

[Jason Louison]

I'll give it a try!

It looks okay, but the spreadsheet can't really handle a function that depends depends on the value of the last function. If I copy the formula down for all the columns, which is the only way to simulate cylinder pressure properly, as the statements are conditional, it results in erratic values or even critical errors.

 

[Jason Louison]

I haven't plotted it because I don't have enough information at this point about the cylinder stroke or valve opening characteristics. The other thing that hasn't really been addressed is air and fuel injection.

What do you have to say about these results?

 

[jack action]

but the spreadsheet can't really handle a function that depends depends on the value of the last function.

I don't see why not. Of course, the first function cannot depend on the last one, but you can compare the results where the answer of the first function must equal the answer of the last function ( ##\frac{x_1}{x_2} = 1## or ##x_1 - x_2 = 0## ).

 

[Jason Louison]

I don't see why not. Of course, the first function cannot depend on the last one, but you can compare the results where the answer of the first function must equal the answer of the last function ( ##\frac{x_1}{x_2} = 1## or ##x_1 - x_2 = 0## ).

I'll just use the equation that depends of volume a the start of the cycle, It seems to work pretty well, but now I need to find out how to simulate valve overlap!