# Automotive Exhaust Gas Pressure/Blowdown Calculations

1. May 7, 2017

### 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

2. May 7, 2017

### 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.

3. May 9, 2017

### Jason Louison

This is whats 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!......?

4. May 9, 2017

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.

5. May 10, 2017

### 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.

6. May 10, 2017

### Jason Louison

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.

7. May 10, 2017

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.

8. May 10, 2017

### Jason Louison

Like this one?!

9. May 10, 2017

### Jason Louison

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

10. May 11, 2017

### Jason Louison

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

11. May 11, 2017

### jack action

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.

12. May 11, 2017

### Jason Louison

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.

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13. May 11, 2017

### jack action

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 $PV = mRT$.

14. May 11, 2017

### Jason Louison

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

15. May 11, 2017

### jack action

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.

16. May 11, 2017

### Jason Louison

I'll give it a try!

17. May 11, 2017

### Jason Louison

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

18. May 11, 2017

### 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.

19. May 12, 2017