Discussion: “Design Improvements to a Biomass Stirling Engine Using Mathematical Analysis and 3D CFD Modeling” (Mahkamov, K., 2006, ASME J. Energy Resour. Technol., 128, pp. 203–215)

For readers wanting to know how computational fluid dynamics (CFD) modeling might help the Stirling engine realize its full potential as a source of sustainable energy, the “closure” from Mahkamov in the December 2007 issue (1), may have proved somewhat disappointing.

The author has chosen for the most part not to answer the specific questions raised by the three discussion contributors (Burton, Burton, and Kinnersly) regarding the large differences between test data and CFD computation (2), the detailed evaluation of Figs. 13 and 14 with regard to entrapment (3), and indeed the confusion arising from some of the figure captions of the original paper (4). Instead Mahkamov has preferred to concentrate on supposed “shortcomings” in the original $γ$ engine, which I designed and which he apparently converted to $α$ mode.

Before discussing in detail these issues, it would be useful to dispel one major misconception. Mahkamov stated (1) that his “calculation (MM) demonstrated that the $α$ machine had higher power output (than the original $γ$ engine) because of its smaller overall dead volume and lower hydraulic losses.”

In converting the original (as designed) gamma engine, with its 180 mm diameter power piston, into an alpha engine, the original 220 mm displacer piston now becomes the power piston, and so the power swept volume increases some 40% (3). As West (5) so clearly explained, “it is the expansion piston that is pushed out during the high pressure phase and can do work.” One does not need CFD modeling to know this will happen; a back-of-envelope calculation (which Mahkamov seemed not to like) will suffice. That the increase in power output in $α$ mode found in both modeling and test work exceeded 40% was due almost entirely to the “gas entrapment problem,” which restricted speed and power in $γ$ mode.

The author has not addressed the most important question raised in discussions of his original paper—the large difference in test-bed power outputs compared with CFD predictions. As I have pointed out (2), the CFD power predictions for the engine in $γ$ mode are around 3.5 times smaller than those observed during test-bed work carried out in my presence, yet Mahkamov’s original claim (6) was that “experimental data, which is available of the mechanical brake power output and speed, indicates that [these are] reasonably close to those predicted by using the 3D CFD model.” Obviously to make such a claim Mahkamov must possess test data and CFD outputs that are reasonably close to one another. We await their publication.

Mahkamov sought to explain away this huge (3.5 times) power difference by stating that the CFD results published in Ref. 6 relate to “a case when the heat flux was at its lowest,” and that the test data collection reported in Ref. 2 were made under “vague” and “uncertain” conditions.

With regard to the original testing of the machine in $γ$ mode (2), one can say the following: Torque was measured by a Heenan and Froude DPX3 dynamometer, calibration of which was corroborated by the earlier use of a Prony brake. Speed was measured by a digital rev-counter from Dynamometer Services Ltd. (Stockend, Bransford, Worcester, Hereford, and Worcester, WR6 5JH, UK).

Contrary to Mahkamov’s claim, the heater was equipped with external thermocouples controlled by a switchable digital readout, and heater temperatures and burner mass flow figures should still both be available from the company’s records. Coolant temperature was measured with a handheld probe thermocouple.

Charge gas pressure was measured by a damped pressure gauge. I can only agree that pressure transducers should have been regarded as an essential part of the test-bed equipment. Among other benefits, they would have made the entrapment problem glaringly obvious at the time, but the company could not afford this extra instrumentation.

With reference to Mahkamov’s claim of excessive charge leak rate, one can only speculate on how the engine came to develop this condition. Is it possible that reassembly of the heater connections after conversion to alpha mode was done by someone unskilled in such matters? This would seem a more likely source of charge-gas leakage than the rocker beam seals. These should not have been disturbed in conversion from gamma to alpha mode.

During the time when I was conducting the first running trials of this one-off prototype, minor rectification was needed to overcome problems caused by incorrect machining; after that only one of the seals leaked significantly. The leak rate was slow enough to permit some 2 h of testing before losing charge pressure of more than 1 bar. If the leak rate during his involvement was really as high as Mahkamov claimed, then it is very difficult to see how the company was able to arrive at the claimed alpha-mode power figure of 7.5 kW (7).

Equally, if the heater tubes were “blocked,” as he stated, then it is difficult to understand how sufficient heat could have been transferred to the charge gas to allow the above power figure of 7.5 kW to be generated.

Burton attempted, in his contribution (3), to estimate the degree to which the entrapment was reducing power outputs in $γ$ mode. He undertook a fairly thorough evaluation of Figs. 13 and 14 of the Mahkamov paper. Mahkamov dismissed these estimates as “back-of-envelope” calculations. We are told that “eradicating the entrapment problem would increase the dead volume to a critical value and therefore these were not modeled.” However, inspection of Fig. 13 of his original paper shows that only for 20 deg either side of top dead center (TDC) was there significant entrapment.

Gapping back this amount represents a dead volume increase of $2.29l/2(1−cos20deg)=0.069l$⁠.

This increase could easily have been offset by using a smaller connecting pipe of 40 mm instead of 50 mm diameter between the two compression spaces.

With regard to the changes of nomenclature and labeling commented upon by Burton (4), Mahkamov stated that “To avoid any confusion in the interpretation of the results, each diagram includes its own caption.” In Fig. 3, the crankshaft angle for the $γ$ engine is zero when the power piston is at TDC. In Fig. 13, for the same $γ$ engine, the crank angle is zero when the power piston is at bottom dead center (BDC). The reader is left to struggle with these changes. As pointed out (4), the captions in Fig. 3 for $Ve$ and $Vc$ are indeed back to front; one only has to look at the values on the vertical volume axis to see that this is so. Moreover, it sounds as though Mahkamov believed that the skewed curve, incorrectly marked $Ve$⁠, is due to biaxial asymmetric drive because he stated, “Fig. 3 demonstrates the variation in the volume of the expansion and compression spaces of the engine with its biaxial asymmetric drive mechanism. As can be seen, asymmetry is apparent in the variation of the expansion space (he meant the compression space) and as such its change results … in minimizing the compression work in the working cycle.”

As close inspection would confirm, the skewed nature of the compression curve in Fig. 3 has nothing to do with biaxial asymmetric drive (this was not present in the engine) but is due to the fact that Mahkamov combined the net effect of two compression spaces, beneath the displacer piston and above the power piston. The skewed curve is due to the 90 deg phase angle between the two pistons.

In responding to Mahkamov’s list of alleged shortcomings in my original $γ$ configuration engine, it is necessary to differentiate between those that were already recognized and in any case needed no modeling to identify (such as the layout of the heater tubes), those where we would agree on key design objectives (such as the imperative to minimize dead space) but disagree over the extent to which this prototype achieved it, and those where the problem described is not intrinsic to the design but arises solely from the mechanical condition of a particular piece of equipment at a particular time.

The problem of “continuous leakage of the working fluid,” discussed earlier, comes into this last category. It is nonsense to imply that seal leakage is an unavoidable characteristic of the design of what he described as the “shafts.” For the benefit of those unfamiliar with this engine, he referred to the rocker beams, which are one of its main innovations (4). It would be a pity if a mistaken notion of insuperable sealing difficulties were to cause other designers to reject the transverse rocker beam as a low-friction way to transmit power from the pressurized envelope of a Stirling engine.

With regard to the “high axial heat conduction losses in the regenerator,” it is not clear whether these were measured losses or were modeled using the assumption that the regenerator film was made of copper (error in original paper) or that it was made of stainless steel (material actually employed in the engine). There should be no need to point out that the 15:1 difference in thermal conductivity between the two metals could introduce a significant error at this point.

While one can only agree with the truism in Mahkamov’s closure, that “the accuracy of CFD modeling depends on the correctness of input data,” it is disappointing that, two years after his original paper, we still do not know the extent to which the errors and misconceptions listed here and in other discussion papers might explain the large disparity between experimental and CFD-modeled power outputs.

Stirling engines have great potential for producing power from a wide range of energy sources including solar, renewables, and waste heat. CFD modeling could help in achieving that potential. By devoting so much space to the perceived shortcomings of an innovative low-budget one-off prototype instead of explaining the apparent shortcomings in his own work, Mahkamov missed an opportunity to show how CFD modeling could help us design better Stirling engines and identify problems we could not diagnose by other means.