Compound steam engines use two or more cylinders to allow the steam to expand, beginning at the input pressure and ending at the exhaust pressure, in two or more stages.  Compounds are referred to by the number of stages of expansions that the steam undergoes.
Most  common is the double compound simple referred to as a compound steam engine.  These consist only of a high pressure and a low pressure cylinder.  Triple or quadruple compounds may have three or four cylinders, respectively, or, to keep the size of the low pressure cylinders practical, two lower pressure cylinders.  Thus a triple expansion engine may have four or five cylinders-one high, one or two intermediate, and two low pressure.
We are familiar with the problems caused by condensation of the steam as it passes from the boiler through the engine.  Condensation takes place as the steam enters the steam chest and cylinder.  Most noticeable is sufficient condensation so as to cause knocking in a cold engine or even a broken crank, connecting rod, or loosening of the bolts holding the supports to the engine base.  These are severe cases, but ones we see far too frequently.
After the engine heats up, condensation of the incoming steam is lessened but continues, because of the fact that as the steam expands during the power stroke, its temperature drops.  It must drop because the expanding steam occupies more space as it expands and the molecules which comprise the steam must now occupy more space, are more widely separated and hence the heat is distributed over more and more volume.  As such, the temperature drops.  If you wish to know, this is an adiabatic expansion, i.e.  expansion to which no heat is added or removed.
So we see that in a simple engine, the temperature range goes from the extreme hot steam (direct from the boiler) to the lower temperature steam which becomes the exhaust.  This lower temperature steam cools the metal of the cylinder preparing the scene for the cold shock to the hot steam entering and resultant condensation of the incoming steam.
On way to avoid the magnitude of this shock is to allow the steam to expand only partially, thus the end temperature will be higher, the metal of the cylinder will remain warmer, and the condensation of the incoming steam will be less.  To continue to derive work from this intermediate steam, it is passed into a next larger cylinder where the expansion is completed and the ultimate (exhaust) temperature reached.
Thus the difference between the input and output temperature of the steam in each cylinder is roughly half that if only a single cylinder is used.  The relative sizes of the pistons are arranged so that each performs approximately one-half of the work.  Triples and Quads are designed to divide the work into three and four equal portions.

A Strath "SWAN" compound steam engine, in-line, with Stephenson valve gear.

Arrangements of cylinders of double compounds determine several types of engines.  Set in-line, aside one another and action on the same crank is a cross compound.  End to end, using the same connecting rod is a tandem compound, which when set vertically becomes a steeple engine.  Cylinders arranged in a vee is termed an angle compound.
In the case of the angle and cross compounds, the pistons are connected to the common crank at 90 degrees so that the crank receives a new power stroke every one-quarter turn.  The tandem or steeple does not enjoy this great advantage that also keeps the crank turning smoothly throughout each revolution.
A pipe betwwen the high and low pressure cylinders carries the intermediate pressure steam.  Frequently, this pipe is enlarged to form a receiver, used to store the steam until it is introduced into the low prssure steam chest.  The receiver may take other forms and may in some cases be omitted entirely.
One can conclude that the compound engine adds a great deal of complication to the single cylinder steam engine and only to reduce the condensation of the steam.  It would seem then that one would be well advised to add insulation and heaters to the steam engine to aid in reducing condensation.
Some engines have been provided with cylinders heaters where steam is introduced to chambers surrounding the cylinders to raise wall temperature to close that of the incoming steam.  One can surmise that this is not practical in the engines that are of interest to us, as the technique is seldom seen.  Steam heaters are more commonly used to heat up the receivers steam.
Insulation, however, is another matter.  Lagging of the cylinders, insulation of the piping, particularly the connecting or receiver piping and that from the boiler will pay off handsomely in improving efficiency.

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