For years, the industry has strived to predict the horsepower consumed by a reciprocating compressor using various equations.?These equations?have become very accurate for the chosen application. In the meantime, the industry?for the most part continues to utilize the theoretical flow equation to predict the flow through the?compressors. The goal here is to begin with the theoretical flow equation, then derive a flow equation that can be used with currently collected compressor horsepower data.

For our purposes here, this author has chosen to use the ratio of specific heats (k) as the main metric. This assumes that the process is adiabatic. In order to prevent confusion concerning isentropic (the entropy of the system remains constant), adiabatic (no heat is transferred to or from the working fluid) and polytropic (real life) processes, it should be recognized that the ratio of specific heats in the following equations can and should be replaced with either the isentropic exponent or the polytropic exponent based on the type of process that is being modeled. Table 1 lists the equation coefficient definitions and descriptions.

Theoretical equation

Since 1988, this author has used only one flow equation. It is a theoretical equation that is described below:

Q = 0.00144 * (Ps/Pb) * (Tb/Ts) * (Zb/Zs) * PD * Evs Eq. 1

This equation is similar to the capacity from cylinder parameters equation presented in the GMRC Report 84-10a. However, over the years, this author has used several different volumetric efficiency equations within the above flow equation. These volumetric efficiency equations were all based on the theoretical volumetric efficiency with various corrections. Below is a list of these equations that were available in a software program that was used:

• Theoretical Equation:

EVs = 1 – CL * (R ^ (1/k) – 1) Eq. 2

• Worthington Equation:

EVs = 1 – 0.01 * R – CL * (R ^ (1/k) – 1) Eq. 3

• Cooper Bessemer Equation:

EVs = 0.97 – 0.008 * R * (Ps/Pb) ^ 0.2 – CL * (R ^ (1/k) – 1) Eq. 4

• NGPSA – Slow Speed Equation:

EVs = 0.96 – 0.01 * R – CL * (R ^ (1/k) – 1) Eq. 5

• NGPSA – High Speed Equation:

EVs = 0.96 – 0.02 * R – CL * (R ^ (1/k) – 1) Eq. 6

Do you notice the similarity? All of the equations contain this relationship:

CL * (R ^ (1/k) – 1) Eq. 7

From this point on, we will call this the clearance term (T). It can be used to simplify the above volumetric efficiency equations as seen below:

• Theoretical Equation:

EVs = 1 – T Eq. 8

• Worthington Equation:

EVs = 1 - T – 0.01 * R Eq. 9

• Cooper Bessemer Equation:

EVs = 0.97 - T – 0.008 * R * (Ps/Pb) ^ 0.2 Eq. 10

• NGPSA – Slow Speed Equation:

EVs = 0.96 – T – 0.01 * R Eq. 11

• NGPSA – High Speed Equation:

EVs = 0.96 – T – 0.02 * R Eq. 12

Using the theoretical equation from above and rearranging:

EVs + T = 1 Eq. 13

This means that if one knows the clearance term, then they should be able to easily calculate the volumetric efficiency; and that the total of both the volumetric efficiency and the clearance term should always be one. These equations are presented graphically in Figures 1 through 6.

Here is where things have stood for a long time for most of the industry. To determine the flow through a compressor, we use the standard flow equation with one of the volumetric efficiency equations. However, over the years these equations have repeatedly been questioned for accuracy. Many people within the industry have used these equations and compared them to the flow measured with a meter. Most of the time they have noticed a significant difference between the two values. Which one is correct? This author would choose the meter over the calculation every time unless there is a concern about the condition of the meter.

Now for a description of how the flow was calculated during these tests. Based on equation 1, one must know the base and suction conditions. In addition, one must be able to calculate the piston displacement using the bore, stroke, rod diameter, and speed. Finally, a person must be able to determine the volumetric efficiency from one of the equations that requires the clearance, ratio of specific heats and ratio of compression. This process is done several different ways depending on the test setup. It seems like a straightforward calculation process – so why the error?

For most testing methods, the volumetric efficiency and the line conditions are not determined at the same points. In other words, the line conditions are determined on the suction and discharge lines leading to the cylinder, and the volumetric efficiency is determined inside of the compressor.

The main point here is that, as an industry, we are calculating flow using either a theoretical flow and volumetric efficiency calculation or using thermodynamics and measured horsepower. Both methods have shortcomings that must be considered. Using theoretical equations will rarely yield the correct values. In general, the theoretical equations will be accurate at only one operating point.

On the other hand, we induce error while trying to calculate the flow using horsepower measured inside of the compressor cylinder along with the thermodynamically determined horsepower per million measured at some convenient location on the piping. Now we can improve the accuracy if one were to collect line conditions closer to the compressor cylinder. The results from the nozzles are more accurate than if we use the loading lines or engine room headers. This author has done this in the past; and seen results from other data sets – and there is an improvement in the accuracy of the flow.

Some organizations use internal cylinder pressures to eliminate the above problems only to introduce other problems. They must apply a pressure drop to the internally determined pressures so that an operator or transducer used by operations can make the measurement. These pressure drops change as the operation of the compressor changes, and must be modeled to accurately determine the internal pressure from a measured line pressure for all operating conditions. However, there is still a difference between the measured flow and the calculated flow.

What else are we missing? Let’s go back to all of the equations and graphs that were previously presented for EVs. Notice that in all of the equations there is a constant offset in the line from theoretical. What happens if the offset is not a constant? Suppose that the offset changes with clearance term. In other words, the sum of clearance term and volumetric efficiency is not always one.

This could be caused by clearance not being used with the same effectiveness across the operating range. Most of us that have been around for a while have seen this occur at extreme operating conditions. If there is too much clearance open on a compressor cylinder end, then the cylinder will start to heat up more than expected. This is due to the excessive amount of gas going in and out of the open unloader pockets during the cycle instead of going out into the discharge line. This is why we have limits when the volumetric efficiency gets too low.

The opposite is true as well. If the flow through the compressor valves becomes too high because there is not enough clearance or the compression ratio is too low, then the volumetric efficiency will suffer. If this is the case, then it is possible that the sum of the clearance term and the volumetric efficiency will not be one.

Some might say that the clearance was not determined (or measured) correctly. It is not uncommon to see that the volume measured by liquid does not correspond to the volume used during compression. By way of analogy, this author has heard others say that it is not correct to determine if a compressor valve will leak by using a water test on the bench at atmospheric pressures and temperatures. If that is true, then why would anyone think that the volume measured in an unloader at room conditions would be accurate for the unloader at all times while operating. The “effective” clearance changes with operating conditions.

Suppose we are able to admit that the volumetric efficiency equations that we have been using for years might need some updating to account for some of these shortcomings. How would you go about determining a new relationship? In the past, this author has performed many horsepower tests and developed lots of equation sets using a modified volumetric efficiency equation. This equation was developed by Panhandle Eastern Pipeline Co. in 1988, and has been previously presented at the Gas Machinery Conference. Here is the form of that equation:

EVs = A + B * T + C * T ^ 2 Eq. 14

Where A, B, and C are numerical constants determined from the test data

Let’s compare this equation to the original set of equations presented earlier: If you assume that the equation is only a straight line, then C will always be zero. A and B can be determined from a simple linear regression of the plotted volumetric efficiency and clearance term data. Below are a few examples of equations developed from test data along with a theoretical equation for comparison.

If this approach is not suitable, then another way to do something similar is to modify the clearance, ratio of compression, or speed to account for the changes in the offset. There are many different implementations of these approaches, and an exhaustive description of them is beyond the scope of this article. Those interested in these different methodologies should consult the HTS Compress Software Documentation for volume correction factor equations. They will yield similar results to the methodology described above. A person can improve the accuracy of the flow calculation by using the volumetric efficiency equation described above or the corrections to clearance, ratio of compression, or speed. How would you implement this with existing data?

If you collected line conditions, then you should input them into an equation of state to determine the enthalpy difference between suction and discharge. This difference can be used to determine the horsepower per million standard cubic feet of gas that is moved by the compressor in a day (HP/MMSCFD):

HP/MMSCFD = (Hd – Hs) * SG / 0.7995 Eq. 15

Divide the measured indicated horsepower by the HP/MMSCFD and the result will be the flow moved by the compressor:

Q = HPm / (HP/MMSCFD) Eq. 16

Plug this flow into the flow equation along with the piston displacement and suction and base conditions, then solve for the volumetric efficiency:

EVs = Q / (0.00144 * (Ps/Pb) * (Tb/Ts) * (Zb/Zs) * PD) Eq. 17

Then, calculate the clearance term using the load step and line conditions:

T = CL * (R ^ (1/k) – 1) Eq. 18

Repeat this for each test condition and plot the volumetric efficiency against the clearance term. Perform a linear regression on the test data to determine the values of A and B in the above equation.

The volumetric efficiency equation developed from a linear regression of the test data collected during a horsepower test will provide a much more accurate flow calculation than is available with existing theoretical equations. If more accuracy is required, then be sure to collect the data that is entered into the equation of state at the nozzles. In addition, be sure to use these same locations when determining the line conditions in normal operations as during testing so that pressure and temperature drops do not cause errors in the flow equation results. Finally, it is possible to utilize the internal pressures with nozzle temperatures, with the provision that pressure drop corrections need to be made for changes in operating conditions such as pressure ratio, flow, and speed.

Ultimately, if you have a flow meter and want to directly match the meter with the equation, then use the flow from the meter instead of the flow calculated from the equation of state when developing the volumetric efficiency against clearance term relationship:

EVs = Qmeter / (0.00144 * (Ps/Pb) * (Tb/Ts) * (Zb/Zs) * PD) Eq. 19

Plot this with the clearance term as described above and perform a linear regression to determine the values for A and B in the EVs equation described above. This will insure that, given similar line conditions, the meter and the equations will match.

Conclusion

While the accuracy of the equations used to predict horsepower has continued to increase, the same cannot be said for the flow equation. It has remained the same for years, even decades. But it is possible to improve the accuracy of the existing flow equation. This can be done with corrections to either the volumetric efficiency equation or to the clearance, speed, and ratio of compression. These corrections improve the results of the flow prediction to match either direct measurements from a meter or a calculation of flow determined from measured horsepower and thermodynamically determined horsepower per million.

References

Panhandle Eastern Pipeline Multistage Compressor Analyzer Software Documentation.

HTS Compress Software Documentation.

GMRC Technical Report 84-10a: Field Measurement Guidelines Compressor Cylinder performance Summary.

Acknowledgment

Based on a paper presented at the Gas machinery Conference, held in Dallas, Texas, October 1-3, 2007.