Compound Expansion

The principles of “Compound expansion” are briefly outlined in the Simple/Compound page of this website which also describes the advantages of compounding as follows:

  • Use of compound expansion allows longer cut-offs to be used, thereby delivering more uniform wheel-rim tractive effort;
  • The ability of compound engines to operate efficiently at longer cut-offs increases their α-coefficient and thus their power-to-weight ratio.
  • The reduction in vibration (or knocking) achieved from the use of longer cut-offs removes the incentive to operate a locomotive with a throttled (partially opened) regulator, thereby allowing full boiler pressure in the steamchest;
  • More uniform torque delivered by compound locomotives reduces the propensity for initiation of wheel-slip at moments of (transient) peak torque.  This renders compound locomotives better suited to heavy haulage;
  • Reduced temperature differentials between steam entering and leaving a cylinder, minimizes heat losses and reduces or eliminates condensation, especially where the low-pressure steam is re-superheated.

Porta expands on these ideas in his paper titled “Fundamentals of the Porta Compound System for Steam Locomotives” as published by Camden Miniature Steam who have kindly given permission for extracts of text and illustrations from the book to be reproduced on this website.

Porta introduces the subject by explaining several important but seldom understood fundamentals that apply to equally to simple expansion.  These fundamentals are outlined on this website in the pages titled:

Porta points out that with compound expansion, steam leakage across valves and pistons will be lower than with simple expansion because pressure differentials are lower.  Also wall effects are less pronounced because cut-offs are longer (giving more uniform cylinder temperatures) and in the case of the low pressure (LP) cylinder(s), pressures, and therefore saturation tempatures) are lower.  However pressure losses are greater because of the the transference of steam from the high to low pressure cylinders, hence the importance of internal streamlining is greater.

The Ideal Compound Expansion EnginePorta offers the following insights into the nature of an ideal compound engine:

  • Equalily of Work in Cylinders: The aim should be to produce an equal amount of work (power) from the high pressure (HP) and low pressure (LP) cylinders.  He illustrates this with two simplified indicator diagrams as below, one describing two or four cylinder machines and the other 3 cylinder machines:

In the case of the two and four cylinder machines where the steam from each HP cylinder exhausts into a LP cylinder, the two parts of the diagram representing the HP and LP cylinder expansion are of equal area, giving a receiver pressure of around 25% of the HP steam inlet pressure.

In the case of the three cylinder machine where the steam from a single HP cylinder exhausts into two LP cylinders, the lower part of the diagram representing the LP cylinder expansion is of twice the area of that of the HP cylinder, giving a receiver pressure of nearer 50% of the HP steam inlet pressure.

  • Resuperheating: Resuperheating of the HP exhaust steam increases the work done in the LP cylinder while at the same time reducing the risk of condensationNote: the receiver pressure must therefore be lower where resuperheating is used, in order to maintain equality of work between the cylinders. The diagram below illustrates the effect of resuperheating.

  • Internal Streamlining:  Internal Streamlining is of great importance in the passages, receiver and resuperheater between the HP and LP cylinders in order to minimize the pressure drop through them and thus maximize the steam pressure entering the LP cylinder(s). He observes also that “what counts is the total pressure drop between the HP cylinder and the LP one, the latter including the triangular loss caused by the LP valve.  Thermodynamics tells us that this type of constant enthalpy transformation is more deleterious the nearer it is from the low temperature sink.  This calls for extra large valves and steam passage areas with a kind of “exaggerated” internal streamlining.”
  • Maximum Superheat Temperature: It is important to maximize the superheat temperature of steam entering the HP cylinder(s), consistent with the tribology employed. Porta recommends HP steam inlet temperature of 450oC, for which purpose, he recommends that his “and NO OTHER” cylinder tribology be adopted.
  • Clearance Volume: Interestingly, Porta recommends a clearance volume of around 16% for the HP cylinder so as to avoid over-expansion at low cut-offs.  This is very much higher than the 6% CV that he recommends for simple expansion locomotives.  Note: Porta mentions that in his own three-cylinder scheme, the clearance volume amounts to 35% because of the higher receiver pressure.  However, for the LP cylinder(s), Porta recommends that Clearance Volume be as low as possible (as for simple expansion cylinders) in order to minimize incomplete expansion losses.
  • High Adiabatic Efficiency in HP Cylinder: Porta notes that being a back-pressure engine, the incomplete expansion in the HP is quite low even when working at long cut-off. The resulting enthalpy increase is, in any case, partially recovered in the LP cylinder.  He notes also that the triangular losses can also be kept small through the use of internal streamlining thus giving very high adiabatic efficiency in the cylinder.
  • Maximum α-Coefficient: Porta strongly recommends against the practice of live steam injection into the LP cylinder, since this will either require that the LP piston and motion be designed for HP steam (thus lowering its α-coefficient).  Instead, Porta recommends the use of an automatic starting valve similar to that used on his 1948 experimental 4-8-0 compound as illustrated below.  The purpose of the valve is to deliver live steam to the low pressure cylinders at reduced pressure when starting, thereby allowing all cylinders to contribute to the initial starting effort at their designated working pressure.

Automatic starting valve used by Porta in his experimental compound 4-8-0 of 1948

The live steam 1 (from the HP steamchest) flow to the receiver 2 until the pressure coming via communication 3, plus the pressure in 2, equals the pressure acting on the underside of the valve 4. The various sections are defined by the pressure ratio P1/P2. The piston 5 has a labyrinth seal.
  • Optimum HP and LP Cylinder Sizes: A secondary argument against the use of live steam injection into the LP cylinder is that it necessitates a reduction in the size of the LP cylinder.  Porta recommends that the LP cylinder be as large as possible in order to minimize incomplete expansion losses, and notes that had Chapelon not allowed 10 bar live steam injection into the LP cylinders of his 140Ps, he would have doubled the LP piston swept volume while using the same motion, hence giving better expansion at high powers, reducing specific steam consumption and thus delivering greater power. Porta recommends that the cylinder diameters and stroke be sized to meet low-speed adhesion requirements.  He further recommends that the HP motion be designed for the pressure differential between boiler and receiver, and that the LP motion be designed for the pressure differential between receiver and exhaust/back-pressure.
  • Determining HP and LP Cylinder Sizes: Porta offers some simple formulae for determining the sizes of HP and LP cylinders based on available adhesion.  He begins with the “standard” TE formula for simple expansion locomotives – viz: where D = wheel dia, p = boiler pressure, d = piston dia, s = stroke and n = number of cylinders.

(Note:  Porta did not include the factor n in his formulae.  It has been added here by the webmaster.)

This equation can be rewritten by introducing π/4 as follows: But T.E. = Adhesive Weight x Friction Coefficient, thus: However π.p.d2/4 = piston thrust P, hence:Whence:where:

    • P = the force in the piston rod in Newtons;
    • μ = the nominal adhesion coefficient (as selected by the locomotive’s designer);
    • W is the adhesive weight in Newtons;
    • D the driving wheel diameter in metres;
    • s the piston stroke in metres; and
    • n the number of cylinders.

Porta then goes on to determine the cylinder/piston diameters for the high and low pressure cylinders in a compound locomotive (dH and dL respectively) as follows:

If ΔpH is the pressure drop between the boiler and the receiver (between the HP and LP cylinders) and if ΔpL is the pressure drop between the receiver and the exhaust (cylinder back pressure), then:

from which [for a compound with nH HP cylinders and nL LP cylinders]


  • Three Cylinder Compounds: Porta makes the following recommendations for 3 cylinder compounds:
    • Higher receiver pressure than for 2 and 4 cylinder compounds (as discussed above);
    • HP cylinder offset to the left-hand side to give a more straightforward front-end arrangement;
    • All cylinders of roughly equal volume but with the LP cylinders being a little larger than the HP cylinder (perhaps to account for pressure drop in the receiver);
    • All three cylinders driving a single axle, with inclined centre cylinder.  Porta claims that he has shown it to be possible to arrange a crankshalf with sufficient strength to support 6000 h.p. or more (4400kW), the stresses being equal to those allowed by AAR for straight axles.

    Porta notes that the design of his 4000 h.p. 2-10-0 for the metre gauge Belgrano Railway in Argentina, the crank angles turned out to be 120 degrees, but that this may not always be the optimum setting.  Note: whilst it would have delivered an uneven exhaust beat, Porta considered that it would not have affected the performance of the boiler because of the use of GPCS.

  • Valve Gear and Cut-off: Porta does not define any firm recommendations for the arrangement of cut-off for the HP and LP cylinders, saying only that cut-offs should be adjusted first by calculation and then verified in operation to equalize the work done in the HP and LP cylinders.  He suggests that if the HP and LP cut-offs are tabulated, their relationship will be approximately linear allowing the adoption of a fixed setting of the valve gear to maintain that relationship.

Several pages of this website include text and diagrams copied from Porta’s “compounding” paper, including the pages covering condensation/wall effects, steam leakage, clearance volume, incomplete expansion and triangular losses.  A more specific reference to his theories on compound expansion can be found on the α Coefficient page.

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.

For a historical perspective on compound locomotives in the UK, an interesting article on the subject published in the July 1992 issue of Steam Classic magazine can be downloaded here.