Predicting engine pressure patterns

Gas machinery operators can use a detailed set of equations to predict the pressure time pattern for a complete cycle of events on an engine. With these pressure time predictions in hand, natural gas pipeline companies can better gauge the operation of their equipment, and optimize their systems by making adjustments as needed.

The process starts with the various physical relationships and operating parameters that must be known, and then covers the development of equations that describe the pressure time relationship throughout the cycle of events. These equations are combined to form a complete pressure time pattern and then the various inputs can be modified to determine changes in the pattern.

Theoretical pressure time trace
This author has often wondered about the possibility of developing a theoretical pressure time trace for an engine; because this is routine for compressors. Over the last two years, we had the opportunity to pursue this possibility.

Initially, we wanted to develop a single equation to model the entire trace. This proved to be very difficult and it could not easily handle changes to the pattern for operational changes. Instead, we decided to break the pattern into small sections and model the equation based on what was occurring at the time in the pattern. This can be done for either a two-stroke or four-stroke engine with relative ease.

Figures 1 and 2 show two pressure time traces for an engine. Both of the traces are the same but the first has top dead center (TDC) located at the left hand side of the plot and the second TDC is located in the middle of the plot. We have chosen to model the second trace with the TDC located in the middle of the plot to simplify the math. Table 1 contains the individual sections of a two-stroke pressure trace. A four-stroke pressure time trace can be divided into similar sections.

Figure 1. A typical pressure time pattern with TDC located at the left on the plot.

Figure 2. A typical pressure time pattern with TDC located at the center on the plot.

Compression
During the compression section of the trace the pressure follows the gas laws. Knowing the initial pressure inside the cylinder is approximately that of the exhaust manifold all other pressures can be found using the relationship:

P1V1n = P2V2n Eq 1

Since we know the initial pressure and volume and final volume, we can rearrange and solve for P2 then the above equation becomes:

P2 = P1V1n / V2n Eqn. 2

For our purposes, we have relabeled this generic equation:

Pc = PEPC VEPCn / Vcn Eqn. 3
Where
Pc = Pressure at the current crank angle
PEPC = Pressure at the exhaust port
covering
VEPC = Volume at the exhaust port
covering
Vc = Volume at the current crank angle
n = Polytropic exponent.
In order to determine the volumes, one must know certain things about the geometry on the unit. The volume in a power cylinder is determined by the following relationship:
Vtot = Vmin + Vstroke Eqn. 4
Where
Vtot = Total volume
Vmin = Minimum volume
Vstroke = Volume between top dead
center and the current piston position.
Vmin = Apiston * (Stroke / CR – 1) Eqn. 5
Where
Vmin = Minimum volume
Apiston = Area of the power piston
Stroke = Length of the stroke
CR = Compression ratio
Vstroke = Apiston * L Eqn. 6
Where
Vstroke = Volume at current power
piston position
Apiston = Area of the power piston
L = Distance of piston from TDC
Apiston = PI * R2 Eqn. 7
Where
Apiston = Area of the power piston
PI = 3.14159
R = Radius of power piston.

These basic equations are used to determine the pressure inside of the power cylinder for every degree of the section from the exhaust port covering until the start of combustion. We do not consider the ignition point as the start of combustion. Instead, we used the change in the first derivative of the pressure trace as the start of combustion. There are various delays between the ignition of the mixture and the effect on the pressure trace due to the ignition source and mixture consistency. It is possible to add these into the equation set as an offset in the start of combustion.

Figure 3. Comparison of model and actual pressure time pattern for a two stroke engine.

The only requirements to determine the pressure for any crankangle in the compression section are the initial pressure at the start of the section (assumed to be the exhaust manifold pressure), the engine geometry (bore, stroke and compression ratio), and the polytropic exponent (assumed to be a constant of 1.43).

Combustion
The initial attempt at the combustion section was a linear relationship between the pressure at the start of combustion and the peak firing pressure. While this looked satisfactory for the combustion section of the pattern, it did not provide a smoothed peak or a gradual start to the combustion process. Mathematically speaking, there were large discontinuities at the boundaries between the sections.

After much consideration (and unsatisfactory trials), we determined to unitize the combustion event as one function. This meant that we would take a typical combustion event and try to make it a length of one unit (from combustion start to peak firing pressure) and a height of one unit (from the pressure determined from the compression section at the start of combustion to peak firing pressure). Unitizing the combustion event allows it to fit all normal combustion events.

The data for a normal combustion event was unitized and then curve fit with several equation types to determine the most accurate. The end result of the curve fit is a fourth order polynomial equation:

CE = -317x4 + 413x3 + 4.5x2 + 0.66x Eqn. 8
Where
CE = Result from unitized combustion
equation
x = (CAc – CAcs) / (CAPFP – CAcs)
CAc = Current crank angle
CAcs = Combustion start crank angle
CAPFP = Peak firing pressure crank angle
The resulting equation for the pressure in the cylinder at any angle during the combustion section is:
Pcomb = (Pmfg + Patm) – Pcomp/100 *
CE + (CAc – CAcs) * FDcs + Pcomp Eqn.9
Where
Pcomb = Pressure in the cylinder
during the combustion section
Pmfg = Manufacturer’s rated peak
firing pressure
Patm = Atmospheric pressure
Pcomp = Pressure at the start of
combustion
CE = Result from unitized combustion
equation
CAc = Current crank angle
CAcs = Combustion start crank angle
FDcs = Rate of pressure rise at the
start of combustion.

Since the equation is unitized, the combustion start and peak firing angles must be known. In addition, the manufacturer’s rated peak firing pressure must be known.

Table 1. Pressure time pattern sections for modeling.

Expansion
Initially, we attempted to use the same set of equations that were discussed in the compression section above. This was close but did not fully match the pattern at either end of the section. As a result of these discrepancies, we modeled a unitary expansion section similar to the method employed for the combustion section described above. The resulting equation is a sixth order polynomial equation:

XE = -13x6 + 45x5 – 60x4 + 37x3 –
8x2 – 2x + 1 Eqn. 10
Where
XE = Result from the unitized
expansion equation
x = (CAc – CAPFP) / (CAEPO – CAPFP)
CAc = Current crank angle
CAPFP = Peak firing pressure crank angle
CAEPO = Exhaust port uncovering
crank angle.
Use the following equation to determine the pressure during the expansion section:
PExp = XE * (PPFP – PEPO) + PEPO Eqn. 11
Where
PExp = Pressure during the expansion
section
XE = Result from the unitized
expansion equation
PPFP = Peak firing pressure from the
end of the combustion section
PEPO = Pressure during the exhaust
port uncovering.

The pressure during exhaust port uncovering is calculated in a similar manner to the equations in the compression section.

PEPO = PPFP * VPFPn / VEPOn Eqn. 12
Where
PEPO = Pressure during the exhaust
port uncovering
PPFP = Peak firing pressure
VPFP = Peak firing pressure volume
VEPO = Exhaust port uncovering volume
n = Polytropic exponent (assumed
to be 1.43)

Exhaust
While this is a short section, and on the pressure trace, it looks pretty straightforward, it ended up being unitized similar to the previous sections. The resulting equation is a third order polynomial equation:

EE = 0.93x3 – 1.18x2 – 0.74x + 1.00 Eqn. 13
Where
EE = Result from unitized exhaust
equation
x = (CAc – CAEPO) / (CAEBD – CAEPO)
CAc = Current crank angle
CAEPO = Exhaust port uncovering
crank angle
CAEBD = Exhaust blow down end
crank angle.

The difficulty in this section is that an additional value must be known for the equation. Most manufacturers provide crank angle for the exhaust port uncovering. In addition to this information, we needed to know the crank angle when the exhaust blow down event is completed. This additional information allows me to unitize the blow down event similar to the previous sections. After the blow down event is completed, the pressure for the remainder of the section is the pressure in the exhaust manifold.

Use the following equation to determine the pressure during the exhaust section between the exhaust port uncovering and the end of the exhaust blow down:

PExh = (PEPO – PEMP) * EE + PEMP Eqn. 14
Where
PExh = Pressure during the exhaust section
PEPO = Pressure at exhaust port uncovering
PEMP = Exhaust manifold pressure
EE = Result from unitized exhaust
equation

Once the current crank angle passes the end of the exhaust blow down, then the pressure remains constant at the exhaust manifold pressure.

Intake
For this section, the pressure must increase from the exhaust manifold pressure to the air manifold pressure. Once this occurs the pressure will remain the same as that in the air manifold. For this section, we simply used a linear relationship between the exhaust and air manifold pressures.

PInt = (PAMP – PEMP)/5 * (CAc –
CAIPO) + PEMP Eqn. 15
Where
PInt = Pressure during the intake section
PAMP = Air manifold pressure
PEMP = Exhaust manifold pressure
CAc = Current crank angle
CAIPO = Intake port uncovering crank
angle

The pressure increases from exhaust manifold pressure until it reaches air manifold pressure and then remains constant at air manifold pressure.

Equilibrium
Similar to the intake section above, the pressure must fall after the intake is covered until it reaches the exhaust manifold pressure. Once the pressure has fallen to the exhaust manifold pressure, it will remain constant for the remainder of the section. We simply used the inverse of the linear relationship developed for the intake section.

PEQ = (PEMP – PAMP)/5 * (CAc –
CAIPC) + PAMP Eqn. 16
Where
PInt = Pressure during the intake section
PAMP = Air manifold pressure
PEMP = Exhaust manifold pressure
CAc = Current crank angle
CAIPC = Intake port covering crank angle

The pressure decreases from air manifold pressure until it reaches exhaust manifold pressure and then remains constant at exhaust manifold pressure.

Pressure time model
Combining all of the above equations for the appropriate section yields a pressure time pattern for the entire cycle. Below are all of the inputs required to develop the equations:

Engine Geometry:
• Compression ratio
• Bore
• Stroke
• Rod length.
Operating crank angles:
• Combustion start
• Peak firing pressure
• Exhaust port uncovering
• Exhaust blow down end
• Intake port uncovering
• Intake port covering (can be determined from the port uncovering)
• Exhaust port covering (can be determined from the port uncovering).
Operating pressures:
• Atmospheric pressure
• Exhaust manifold pressure
• Air manifold pressure
• Manufacturers rated peak firing pressure
• Polytropic exponent for compression
• Polytropic exponent for expansion.

Utilizing the above inputs with the equations in the previous sections provides a complete pattern. At the end of this article is a comparison of the model and an actual pressure time pattern.

Discussion of comparison
It is possible to see from Figure 3 that the modeled pattern closely resembles the actual pattern. Since actual data for the pressure trace pattern was available, several of the inputs were slightly modified so that the model would come closer to the actual pattern. The primary changes were to the polytropic exponents and the compression ratio. Once these changes are complete, then the model can be used to do what if scenarios for changes in the other input parameters. The shortcomings of the model appear to be in the compression and expansion sections of the model. Additional work in these areas can minimize or eliminate these differences.

Conclusions
Below are some of the conclusions that can be made concerning the pressure time model:

1. The pressure time patterns developed from the model can closely fit actual data.
2. It does not require actual pressure data to develop the model.
3. It can be used to predict what will happen to the pressure pattern if one or more operating parameters are changed.
4. There are possible improvements that can be made to the accuracy of the pattern in the compression and expansion sections of the pattern.
5. Future development could address ignition system delays instead of using a combustion start angle.
6. A four-stroke model can be developed using a similar process.

Acknowledgment
Based on a paper presented at the Gas Machinery Conference held in Albuquerque, New Mexico.

The author
Manny Angulo is Senior Engineer for Anderson Consulting Training & Testing, based in Houston, Texas.