I have spent an inordinately long period of time resolving the intricacies in combustor diffuser. Considering the lower temperatures we are dealing with, pressure losses can be rather devastating. As a result, I looked at a myriad of texts and methods to develop our pressure model for our combustion system. One pair of texts written by Jack D. Mattingly have me ready for therapy.

## Introduction

There are not many books that do a good job detailing how combustors perform. Arthur Lefebvre and Dillip Ballal wrote *Gas Turbine Combustion*, which is the be-all, get-all book on the combustor. That book is theory based, with a ton of references to research and testing, but no applied examples. Mattingly wrote *Elements of Propulsion* and co-authored *Aircraft Engine Design* with Heiser and Pratt. Both texts have been helpful in many areas of research and I use them constantly as methods of cross-reference and validation. I want to focus this discourse on the excellent combustor diffuser reference in the latter, co-authored book. In particular, I want to prove that Mattingly is actually trying to make me crazy.

The purpose of diffusion is to reduce the velocity of a fluid while maintaining the pressure. The combustion diffuser does this by raising the Static Pressure of the fluid, and simultaneously minimizing its Total Pressure Loss. At least, that’s the dream.

The following section will attempt to step through the basic mathematics of such diffusion, with an example to show where my concern lies. It is not intended to be a reference, and the studies behind the statements are completely neglected for brevity. The following sample is restricted to the basic, 9° angled-wall diffuser with splitter vanes, and assumed inefficiencies for simplicity sake. The list of variables and other notations are at the bottom.

## Existing and Proposed Conditions

Here we start out with standard safe values. There is really only one variable that a designer has to work with for diffusion, namely AR. Effectiveness (a pseudo efficiency) is determined by the materials and the conditions that are present in the design. Other factors are important, but have negligible bearing on the exit velocity. These are not germane to this review. Note: effectiveness of 0.9 was selected, being the same as Mattingly used in one of his examples.

## Coefficient of Pressure Recovery

Examination of this shows how static pressure behaves in relation to a change in area and the effectiveness of the diffuser. This is set in relation to the entry Dynamic Pressure.

The Static Pressure rise can then be determined. This is straight forward, and is consistent throughout all texts.

## Total Pressure Loss Coefficient

This is where things get a bit sticky. Notice that this equation indicates the behavior of the Total Pressure in the diffuser. It is also set relative to the initial Dynamic Pressure.

At first glance it appears as though everything is fine. We have Static and Total Pressure leaving the unit. In the following section “Total Pressure Ratio”, we will begin to see where things don’t line up. I wanted to mention that Lefebvre and Ballal discuss the Total Pressure Loss Coefficient, but leave the reader feeling that the term is usually derived from detailed measurements. The term has little citation in other texts for predicting the Total Pressure outcome. In fact, Mattingly never even mentioned it in his solo-book, yet Mattingly, Heiser, and Pratt used it everywhere.

## Total Pressure Ratio

The following is the Total Pressure Ratio that is furnished by Mattingly, Heiser, and Pratt. Inspection of the balance check should indicate that the resulting Total Pressure does not correspond with the Total Pressure determined from the Total Pressure Loss Coefficient in the last section. When I first started this study years ago, I thought I was doing something wrong. Nothing would line up, but I kept going back because the text was rather detailed.

I discovered the following relationship noted in Mattingly’s solo book. After I plugged it into the model, the difference was night and day! If you carefully inspect the following, you should see the Total Pressure Loss Coefficient in the numerator of this version as well as the previous.

Everything checked through, and every glitch was solved!

I would bet that this version was derived directly from the P/P0 relationship with Mach. I began to wonder if the Total Pressure Loss Coefficient in the co-authored book was a mis-print. Mattingly stated that the exiting fluid Mach value needed to be maintained below Mach 0.10 in order that the Total Pressure Ratio remain above 0.975, a reasonable trade-off between performance and friction. So I set out to ensure that was true, and to see how that affected our model. I was so happy that I found the missing link.

That feeling did not last long.

## Mach and Mattingly’s Recommendation

Now that we have Static and Total pressure, Mach can be determined. So let’s plug in Mattingly’s relationship, and see what shakes out. Mattinglys’ P02/P01=0.9846, so we should have a Mach value somewhere below 0.1, right? Umm..No.

It would appear, even after long-hand verification, that the Mach of the fluid will only be reduced to 0.18. This seems odd, and a bit fast. So after rechecking and altering… and screaming, I tried throwing insane AR values at it, like AR=100. The result?

Nothing. The Mach curve asymptotes near 0.17. There is no way to get below this value. I wanted to prove this to myself. So I reversed the equation, and developed the effectiveness curve as a function of AR, based on any P/P0 ratio given for Mach 0.1. NASA’s Isentropic Flow equations verified a ratio of 0.9931 for M=0.1.

Note: ignore the top variable, delta P/delta P0. I learned some interesting things there as well, but beyond the scope of this discussion.

You “ain’t” never, never, ever-ever-ever gonna get that Total Pressure Ratio equation to work. Well, not until they develop that super ceramic that has inverted effectiveness enhancement properties. I am not sure why this fails, save for the fact that I have beaten it to death. Moreover, I’m so frustrated with it, that I felt the need to document it here.

## Conclusion

There is some confusion presented by these authors. Once we understood it, it did not take us long to validate our models elsewhere. The Co-authored text is good. I wish to point out that the writers of *Aircraft Engine Design* left a significant note, in a rather muted manner. They stated that these relationships are individually significant to which specific results you are focused on.

Nothing in this discussion should be construed to say that I do not like or recommend Mattingly’s books. Emphatically quite the contrary. I have read his books repeatedly (wow, that sounds sad, now that I hear it said out loud). But really, these are great books, and I continue to refer to them regularly. I simply found this particular difference in statements rather frustrating, especially when the same author was involved in both texts. I’d love to meet this amazingly intelligent fellow (and a great writer). However, when I do, I might have to ask, “What gives?”

## Variables

AR = Area ratios of entry and exit planes, A2/A1

γ = Heat Capacity Ratio

P0 = Total Pressure in kPa

P = Static Pressure in kPa

η = effectiveness

M = Mach

Subscripts

1 = entry values

2 = exit values

D = Diffuser

## Bibliography

Lefebvre, Arthur H., and Dilip R. Ballal. *Gas Turbine Combustion.* 3^{rd} ed., CRC Press, Taylor & Francis Group, LLC, 2010.

Mattingly, Jack D. *Elements of Propulsion: Gas Turbines and Rockets.* 2^{nd} ed., AIAA Education Series, American Institute of Aeronautics and Astronautics, Inc., 2006.

Mattingly, Jack D., William H. Heiser, and David T. Pratt. *Aircraft Engine Design.* 2^{nd} ed., AIAA Education Series, American Institute of Aeronautics and Astronautics, Inc., 2002.