Thread: 4BD1T Turbo Sizing and Performance Prediction.

Q. Using this information is it possible to determine a theoretical fuel consumption rate? Happy to list some assumptions to simplify things, such as constant speed of 100km/h (zero acceleration) and zero wind speed at 0km/h.

Originally Posted by Judo

Q. Using this information is it possible to determine a theoretical fuel consumption rate? Happy to list some assumptions to simplify things, such as constant speed of 100km/h (zero acceleration) and zero wind speed at 0km/h.

The biggest factor in fuel consumption is engine load, which is your engine.
Some coast-down tests with a bit of data (weight etc) can give you a rough idea of the power required for steady state cruising.

The other part is estimating or calculating engine efficiency at part load. That is the hard part.

Nevermind. Found it.

So far we have determined the density ratio (DR) for the air mass flow needed for effective combustion of the fuel required for the desired performance, and made an estimate of the pressure ratio (PR) to achieve that density ratio. That estimate was used to read the approximate adiabatic efficiency from a compressor map.


Now we want to determine the required PR to achieve the DR.


Recall for the compressor, density ratio was defined as DR = density of air at outlet / density of air at inlet


If an intercooler is used then use density at intercooler outlet in place of compressor outlet.


Also recall that PR = absolute pressure at outlet / absolute pressure at inlet


In the calculation for DR we use the following formula for both the outlet and inlet density:

Density of air = (Pa x M) / (R x Ta)

Where:
Pa is absolute pressure in Pa (Pascal)
M is molar mass of air = 0.0289644 kg/mol
R is ideal gas constant = 8.31447 J/mol K
Ta is absolute temperature in degrees K (Kelvin) = degrees Celcius + 273

Then, both M and R are cancelled out and density ratio simplifies to:

DR = POUT / PIN x TIN / TOUT

Where:
POUT is absolute pressure at compressor, or intercooler outlet
PIN is absolute pressure at compressor inlet
TIN is absolute temperature at compressor inlet
TOUT is absolute temperature at compressor, or intercooler outlet

Rearranging to find PR we get:

PR = POUT / PIN = DR x TOUT / TIN

Now the difficulty that arises is we need to know the PR before we can find the outlet temperature

TOUT = (TIN x (PR0.288 - 1) / adiabatic efficiency) + TATM

where:
TOUT is the outlet temperature (C)
TIN is the absolute inlet temperature (K) = TATM + 273.15
PR is the pressure ratio developed by the compressor
0.288 is {(k-1) / k} where k is the ratio of specific heats of dry air
adiabatic efficiency is the efficiency found from the compressor map
TATM is the local ambient temperature ( C)

To use the methods that Garrett and other some others give for selecting a turbo they assume we know TOUT but in reality the best that can be done is make an educated guess.


We could use the outlet temperature we found in the last stage in the equation given above for PR. Then repeat the process if the calculated PR differed too far from the approximate PR used to find the adiabatic efficiency and outlet temperature.


In the equation for outlet temperature the temperature increase created by the compressor is the left part of the expression:

(TIN x (PR0.288 - 1) / adiabatic efficiency)

And for the intercooler:

TMAN = TOUT – [(TOUT – TATM) x effectiveness]

where:
TMAN is the inlet manifold temperature (C)
TOUT is the outlet temperature from the compressor (C)
TATM is the local ambient temperature ( C)
effectiveness is the intercooler effectiveness (usually between 0.6 and 0.7)

Here the temperature reduction created by the intercooler is the left part of the expression:

[(TOUT – TATM) x effectiveness]

There is another way!
It is easy to find DR for a given PR if we know the adiabatic efficiency of the compressor and the effectiveness of the intercooler (if fitted). This is useful in a spreadsheet for our calculations, by allowing us is to construct a 'look-up' table (or graph) of DR vs PR


In the look-up table, we can have a row (or column) for:

PR over the range we might be interested in
TOUT (from the equation above) for each PR value
TMAN (from the equation above) for each TOUT value
DR from DR = PR x TIN / TOUT if no intercooler
DR from DR = PR x TIN / TMAN if an intercooler is used

Then it is simply a matter of finding the required DR (intercooled or non-intercooled) in the table and looking up the corresponding PR.


Then plot the points for PR vs air flow on the compressor map to find the adiabatic efficiency. If the new adiabatic efficiency is close to the value we used above, then we have a valid PR.


If it is too far out, we need to use the new adiabatic efficiency to re-calculate the look-up table.


Once we have the above values, the next stage is to determine the power needed to drive the compressor and move on to the turbine.


I should point out here, that most methods for choosing a turbo take a different approach than I have followed in this thread, and start with a target boost pressure at each engine rpm of interest. Then the sequence of calculation becomes:

PR
Air flow
Adiabatic Efficiency
Density
Air mass flow
Fuel rate from A/F ratio
Power from fuel rate and SFC

If the target power is not achieved, then guess another boost pressure and repeat the process.


This procedure is useful if you have already installed your turbocharger and can monitor the boost pressure at the different rpm points. It allows you to see the approximate power and conduct 'what if' calculations to see what can happen if the boost pressure is increased or decreased.


However remember that with a diesel engine the fuel rate is not going to change, simply because you have 'such and such' an air flow or boost pressure. Governor adjustment is required, and the full load adjustment will be for a particular rpm point. With our mechanical injection pump we don't have the ability to MAP the fuel rate to air flow over the full range of engine revs.


The best we can do is, over the range of rpm and load, use a setting that doesn't create smoke or egt's that are too high. So at some sections of the range, we may have surplus air, but we can't increase the fuel rate (torque/power) there because of consequences elsewhere. I will discuss this further in another post.

For our examples at 3000 rpm, we arrived at the following requirements for density ratio:


(Ex a) for 90 kW:
DR = 0.108 kg/sec / 0.0887 kg/sec = 1.218 (for A/F = 18:1)
DR = 0.120 kg/sec / 0.0887 kg/sec = 1.353 (for A/F = 20:1)


(Ex b) for 135 kW:
DR = 0.162 kg/sec / 0.0887 kg/sec = 1.826 (for A/F = 18:1)
DR = 0.180 kg/sec / 0.0887 kg/sec = 2.029 (for A/F = 20:1)


(Ex c) for 180 kW:
DR = 0.216 kg/sec / 0.0887 kg/sec = 2.435 (for A/F = 18:1)
DR = 0.240 kg/sec / 0.0887 kg/sec = 2.706 (for A/F = 20:1)


And for these examples we estimated an adiabatic efficiency of:


(Ex a) for 90 kW:
adiabatic efficiency = 0.70 (for A/F = 18:1)
adiabatic efficiency = 0.72 (for A/F = 20:1)


(Ex b) for 135 kW:
adiabatic efficiency = 0.74 (for A/F = 18:1)
adiabatic efficiency = 0.74 (for A/F = 20:1)


(Ex c) for 180 kW:
adiabatic efficiency = 0.72 (for A/F = 18:1)
adiabatic efficiency = 0.72 (for A/F = 20:1)


Also for these examples we used a local ambient air temperature of 303 K (approx 30 C) and an intercooler effectiveness of 0.65


Using the look up table in a spreadsheet the following PR's are found:


(Ex a) for 90 kW (no intercooler):
PR = 1.39 (for A/F = 18:1)
PR = 1.64 (for A/F = 20:1)


(Ex b) for 135 kW (with intercooler):
PR = 2.02 (for A/F = 18:1)
PR = 2.29 (for A/F = 20:1)


(Ex c) for 180 kW (with intercooler):
PR = 2.85 (for A/F = 18:1)
PR = 3.24 (for A/F = 20:1)


And the greatest difference in adiabatic efficiencies was for (Ex a) and A/F = 18.1 where the new adiabatic efficiency is 0.68 (was 0.70). This changes the PR to 1.4 from 1.39.


I will make the spreadsheet available later.

Specific Work
Is the work per unit mass evaluated for conventional turbo machinery such as pumps compressors and turbines. The theory for specific work for compressible fluids is useful for matching a turbocharger's compressor and turbine.


The work from the turbine = work from compressor + turbo losses (friction).


Specific work has SI units:
N m/kg = J/kg = m2/s2
N (Newton) is the derived SI unit for force (kg m/s2)
J (Joule) is the derived SI unit for work and energy (N m)

For Isentropic Compression(or Expansion)
p1 v1κ = p2 v2κ
where:
p = absolute pressure (Pa)
v = volume (m3)
κ = cp / cv = ratio of specific heats (J/kg K)
cp = specific heat at constant pressure
cv = specific heat at constant volume
K (Kelvin) is SI unit for absolute temperature

Specific Work of Compressor
A compressor increases the air pressure from p1to p2 and the specific work can be expressed as:
w = κ / (κ -1) x R x T1 x [( p2 / p1)^((κ-1)/κ) - 1]
where:
R = individual gas constant (J/kg K)
T = absolute temperature (K)

Specific Work of Turbine
A turbine expands the exhaust gas from p1 to p2 and the specific work can be expressed as:
w = κ / (κ -1) x R x T1 x [1 - ( p2 / p1)^((κ-1)/κ)]

Power from Specific Work
Power = specific work x mass flow

Head
With turbo machines it can be convenient to express specific work in terms of head (of the fluid).
w = g h
where
h = head (m)
g = acceleration of gravity = 9.81 (m/s2) approximately
Then head is given by:
h = w / g

Just a quick clarification for anyone going through the equations - in the equation given for temperature increase during compression

i.e. TOUT = (TIN x (PR0.288 - 1) / adiabatic efficiency) + TATM

PR0.288 should say PR^0.288 (or PR to the power of 0.288).

HTH

1/3.5 is a quick and easy way to get that expression to whatever accuracy you require.

(1.4-1)/1.4 = 1/3.5 = 0.2857142..........