Modeling Turbo RPM for Desired Pressure Ratio in BMW

[bm0p700f]

I am trying to model the RPM at which the turbo I plan to fit to my BMW will develop the desired pressure ratio.
I have determined air flow into the engine at the desired pressure ratio and have determined the rpm at which peak power and torque should arrive. I have also determined the size of these quantities.
I have used a basic Otto cycle model to determine exhaust temperatures. However the model does not factor in heat loss and assume instantaneous combustion at TDC. It also does not factor in the early opening of the exhaust port before BDC. I would like to factor in heat loss and burn fraction as function of crank angle but the equation realting cylinder pressure to crank angle, incorporation the burn fraction using the wiebe function, is a bit complex. Also as pressure appears on both side I am not so sure solving it is going to help me much apart from give me a headache.
Onto the turbine/compressor. Knowing the desired pressure ratio I have determined the work required to compress the air in the inlet pipe to the intercooler. From this I have determined the turbine pressure ratio.
The equations I have used are;
Cp= (dm/dt)*cp * Tinlet * (CPR ^((gam - 1) / gam) - 1) / nc
Source http://www.grc.nasa.gov/WWW/K-12/airplane/ctmatch.html
Cp is compressor power (W)
Dm/dt is mass flow rate of air through compressor (kg/s)
Cp is the specific heat at (constant pressure?) This seems wrong as pressure changes anyway cp 1108 J/kg/K
Tinlet is the ambient air temperature 293K.
CPR is the pressure ratio = 1.62
Gam is the ratio of specific heats = 1.4
Nc is the compressor efficiency ~0.7
Tp = (dm/dt)*nt * cp * Tex * (1 - TPR ^ ((gam -1) / gam))
Then;
TPR ^ ((gam -1) / gam) = 1 - Tt2 * (CPR ^((gam -1) / gam) - 1) / (nc * nt * Tt4)
Tp = turbine power (W)
Tex is exhaust temperature
Nt is turbine efficiency
The basic model predicts an exhaust temp of 1024 K but this does not vary which is a problem. To provide a pressure ratio of 1.62 in the inlet manifold a turbine pressure ratio of 0.73 is required with a turbine efficiency of 0.6929.
However no of this work has allowed me to have any insight into what RPM I will see my desired pressure ratio and what rpm boost pressure will start to build.

Can anyone help me out here.

 

[jack action]

You need to learn about compressor map to find compressor RPM vs mass flow rate and pressure ratio. This is a http://www.turbobygarrett.com/turbobygarrett/tech_center/turbo_tech103.html" for that.

 

[bm0p700f]

You miss understand. I have sized the turbo using compressor maps. I understand how to do that! What I now need to do is determine the engine RPM at which I will achieve the desired pressure ratio. A compressor map cannot do this. If you read the the link you posted you will see this.

A turbine map will only show me how pressure ratio changes with gas flow into the trubine. That is what I need but the turbine maps for the turbo I have chosen are not available.

I therefore need to model how exhaust temperature and pressure changes with engine RPM so can determine at what engine RPM a pressure ratio of 1.62 in the compressor will be achieved. The only way I can see how to do this is to determine the work done by the compressor in pressuring the air and taking that as the work done by the exhaust gas on the turbine. From that I have already tried determined the pressure ratio acorss the turbine. What remains unknown to me at the moment is the exhaust manifold pressure (against engine speed). If I knew this then I can work out what I want to this is what I need a bit of help with.

 

[jack action]

Sorry, I thought you wanted to know the compressor's RPM.

If you have determined the airflow, as you said in the OP, then you have the answer to your question:

volumetric flow rate [m³/s] = volumetric efficiency X engine displacement [m³] X RPM [rad/s] / (4*pi)

mass flow rate [kg/s] = atm air density [kg/m³] X volumetric flow rate [m³/s]

atm air density = 1.225 kg/m³

So (in SI units):

RPM = (4*pi) X mass flow rate / atm air density / engine displacement / volumetric efficiency

Read http://www.epi-eng.com/piston_engine_technology/volumetric_efficiency.htm" for more info.

The mass airflow through your compressor, engine or turbine is the same (assuming no bypass).

 

[jack action]

I just realized that I forgot something about volumetric efficiency (VE) (which is not mention in the link).

For engine with forced induction, you have to multiply the original VE by a "compressing factor" which, for a typical street car is between 1.4 and 1.5. You can find that factor based on an adiabatic compression this way:

$$VE_{boost} = VE_{un-boost} \left( \frac{boost\ gauge\ pressure}{atm\ pressure} + 1 \right) ^{1/{1.4}}$$

Or:

$$VE_{boost} = VE_{un-boost} P_r^{1/{1.4}}$$

Where $$P_r$$ is the compressor's pressure ratio.

 

[Herr-Kuhn]

You could try to calculate this, but you might be better off in selecting a turbocharger that you know will work as desired for the max output you seek...and if it doesn't perform as you so desire, simply change the A/R on the housing. I think you will struggle to get a definite answer on when it will light up.

There are a lot of variables...engine size, compression ratio, barometric pressure, altitude (same as barometric basically), ignition advance, fuel octane...determines how much advance you can run, port design, manifold design, heat retention in the manifold, etc. The size of the manifold and the length of the runners and how they all collect plays a big role in how the turbo will respond. Remember, you are talking about transient response here and there are an enormous amount of variables at play. This doesn't even begin to talk about the cold side pipes and intercooler volume...though most turbos fill this volume very quickly.

I'm assuming you are talking about a bimmer inline 6 here. Something in the GT35 or larger would be a good starting point. I'd be interested in learning more about your project. I build forced induction systems for European cars.

If you pick a reasonably well sized and matched turbo there should be no reason to not expect to have full boost by say, 3,500 RPM in a higher gear. Don't forget that the gear ratios, weight of the car and aerodynamic loading also play a huge role in how it will build boost. You will never get the same transient response in 1st gear that you will in say 3rd or 4th gears.

What you're trying to accomplish on paper is likely possible to predict but it will take an extreme amount of time to do so and be prepared to be changing all sorts of variables in your model in an attempt to get a range.