Incomplete Expansion

Incomplete Expansion of Steam in the Cylinder

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The term “incomplete expansion” is used to describe the curtailment of the expansion of steam inside a locomotive’s cylinder brought about by the opening of the exhaust valve, thus allowing the remaining energy in the steam to be lost to atmosphere. They may also be described as losses caused by insufficient cylinder volume. Either way, the effect is to limit the work done by each piston stroke which in turn inhibits the locomotive’s power output and efficiency.

Incomplete expansion losses are best visualized diagramatically as shown on the indicator diagram below, copied from Porta’s “Compounding” paper as published in Camden’s book “Advanced Steam Locomotive Development – Three Technical Papers“. The incomplete expansion loss is shown as the hatched area ‘K’.

In his “Compounding” paper, Porta goes on to state:

The loss of the single expansion engine is the incomplete expansion. This loss is heavier the higher the boiler pressure is, indicating the benefits of compounding.

Incomplete expansion also increases with longer cut-off (i.e. fatter indicator diagram) which is why short cut-off working is held to be the secret of economical operation. In fact this is not so, since short cut-off working results in greater cooling of the cylinder walls and a consequential increase the likelihood of condensation as explained in the Condensation page of this website. Porta therefore recommends a cut-off of 20 to 25% for single expansion operation as being a compromise between the conflicting aims of minimizing condensation and minimizing incomplete expansion. It further explains why he recommends compound expansion since this allows the use of much longer cut-offs, resuperheating to prevent all possibility of condensation, and expansion losses limited to the low pressure cylinder where their effects are less pronounced.

Note: Incomplete expansion losses are also critically affected by Clearance Volume as explained on the Clearance Volume page of this website. Porta also points out that incomplete expansion losses are affected by condensation, increasing as wall effects increase and vice versa.


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 triangular losses 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.