Triangular Losses

Triangular Losses in Cylinders

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The term “triangular losses” is used to describe the rounding of the corners of a locomotive’s indicator diagram caused by the opening and closing of valves, and whose effect is to reduce the area of the diagram and thus the work done by each piston stroke which in turn reduces power output and efficiency.  Some of these do not represent losses of energy so much as “losses of potential”.

Triangular losses are perhaps best described in Porta’s “compounding” paper published in Camden’s book “Advanced Steam Locomotive Development – Three Technical Papers” and consist of the rounded corners of a typical Indicator Diagram as compared to an “Ideal” diagram, as shown on Figure 1 of Porta’s paper as shown on the Condensation and Incomplete Expansion pages of this website.

A simplied indicator diagram is shown below to illustrate triangular losses.

Three loss areas are shown:

  • Area A occurs during the steam admission phase where throttling occurs due to the narrowing of the admission port as the valve approaches cut-off;
  • Area B occurs when the exhaust port opens before the piston reaches the end of its stroke, allowing the escape of steam before it fully expands into the cylinder;
  • Area C is the compression (or pre-compression) that occurs when the exhaust valve closes before the piston reaches the end of its stroke.

It might be argued that Areas A and C should not be regarded as “losses” per se. Triangle A primarily represents lost potential by virtue of steam that failed to enter the cylinder because of throttling during valve closure. (An entropy rise results from the throttling process so some energy loss is involved also).

Triangle C also largely represents lost potential since it is apparent that the area inside the diagram would be larger (and therefore more power gained) if the exhaust port were to stay open until the end of the piston stroke.  However, since the steam is compressed elastically it returns most of the energy that it absorbs during the reverse stroke, some being lost through an increase in entropy.

Notwithstanding, pre-compression offers two advantages – (a) it cushions the piston’s inertial (deceleration) forces that would otherwise have to be resisted by the connecting rod and crank pin; and (b) it builds up the cylinder pressure prior to admission and thus helps to reduce or eliminate another triangular loss that would otherwise arise at the top corner of the diagram due to a delayed rise of pressure as steam flows into the empty cylinder when the inlet valve opens.

In fact triangular losses are more complex than shown as ‘A’ in the simplified diagram above. Porta draws attention to the triangular losses that actually occur at the start of admission (as the inlet valve opens) and illustrates his point in Fig 8 of his “Compounding” paper as below:

Note: The “net definite pre-admission that Porta refers to can also be referred to as “lead” as defined in the Valves and Valve Gear page of this website. What he is saying here is that the use of lead causes a small triangular loss (shown with horizontal shading).  On the other hand, absense of lead (or inadequate lead) results in a much larger triangular loss (shown with vertical shading).

Porta goes on to point out that the admission losses should actually include the area under the “Nominal Steam Pressure” line (indicated in yellow shading below), demonstrating the importance of (a) operating at maximum boiler pressure; (b) the use of large steam pipes, large steam chest and internal streamlining to minimize the pressure drop between boiler and cylinder.


Triangular losses cannot be eliminated, but they can be minimized by careful design – for instance:

  • use of large valves and port openings to reduce steam velocity and consequent flow losses;
  • optimizing valve events using computer simulation such as those of Prof Bill Hall and Dr Allan Wallace;
  • in the case of piston valves, by the use of long-travel valves that pass over the ports at higher speed, thereby reducing the time that the entry or exit of steam is throttled through a partialy opened port.
  • use large steamchests to minimise steamchest pressure drop during admission – ideally steamchest voiume should equal cylinder volume.

Note: “Losses of potential” (as described above) are real losses in the form of wasted capital rather than wasted energy.  Minimizing losses in potential increases a locomotive’s performance and therefore its return on capital.


Porta’s paper titled “Fundamentals of the Porta Compounding System for Steam Locomotives” addresses other associated factors that detract from a locomotive’s cylinder efficiency, including condensation/wall effects, steam leakage, clearance volume and incomplete expansion as described elsewhere on this website.  More specific references to his theories on compound expansion can be found on the α Coefficient and Compound Expansion pages.

Sincere thanks to Adam Harris of Camden Miniature Steam, publishers of “Advanced Steam Locomotive Development – Three Technical Papers” for allowing the sections of the book to be published on this website.