Technical Terms
Efficiency and Losses
Thermal Efficiency
The “thermal efficiency” of any engine is defined as the amount of useful energy output divided by the amount of energy input . It is not a fixed quantum but varies according to the engine’s load and conditions of operation.
In the case of steam locomotives, the term thermal efficiency may refer to “Cylinder or Indicated Efficiency“, “Drawbar Thermal Efficiency” or even “Boiler Efficiency“. These are described on separate pages, however their definitions are importantly different as outlined below: Three types of efficiency are described on separate pages as follows:
 Cylinder efficiency is defined as the the amount of energy delivered by the cylinder to the piston divided by the amount of energy delivered to the cylinder in the form of steam delivered to the steamchest;
 Drawbar efficiency is defined as the the amount of energy delivered at the locomotive’s drawbar (the hook at the back of its tender) divided by the amount of energy available in the fuel placed into its firebox.
 Boiler efficiency can be defined as the amount of energy delivered from the boiler in the form of steam divided by the amount of energy delivered to the firebox in the form of fuel/chemical energy.
Both the cylinder and drawbar efficiencies vary with speed and power output, maximum cylinder efficiency being achieved at much higher speed than maximum drawbar efficiency. This is because, as speed rises, a locomotive’s rolling resistance also rises and the tractive force avalable at its drawbar falls until, at a certain speed, the drawbar force becomes zero and thus the drawbar efficiency also becomes zero.
Cylinder efficiency
Cylinder efficiency is governed by the shape of the Indicator Diagram and in particular by the losses that are evidenced by it – most especially expansion losses, condensation losses and leakage losses.
It should be noted however, that even without these losses, cylinder efficiency is limited by Carnot’s equation which states that the maximum theoretical efficiency of any heat engine is governed by the temperature difference between its heat source and its heat sink. See Thermodynamics page for further explanation.
Note: Isentropic efficiency is another (but very different) measure of cylinder efficiency. Instead of describing the ratio of work output to work output, it describes the ratio of work output with maximum possible work output based on steam conditions – see Thermodynamics definitions.
Drawbar efficiency can be seen as the sum of the efficiencies of a locomotive’s various components. Wardale provides examples of these in his book “The Red Devil and Other Tales from the Age of Steam” where (in Table 78, page 457) he quotes figures for standard and (proposed) modified Chinese Class QJ locomotives, and where (on page 501) he suggests what might be achieved from the further development to the level of “Third Generation Steam” traction.
The figures from these pages are combined in a single table below, however it is recommended that the qualifying texts from both Table 78 (page 457) and page 501 of Wardale’s book be read in association with them.
Item  Standard QJ  Modified QJ  Third Generation Steam 
Boiler combustion efficiency  78%  87%  95% 
Boiler absorption efficiency  78.2%  80%  90% 
Auxiliary efficiency factor  93.1%  94%  96% 
Cylinder efficiency  16.4%  19.05%  22% 
Transmission efficiency  89%  93%  94% 
Drawbar efficiency  94%  95%  96% 
Overall drawbar thermal efficiency = product of all the above 
7.8%  11.0%  16.3%* 
Maximum drawbar thermal efficiency is usually reached at modest speed and power outputs such that increasing rolling resistance and increasing fuel carryover (in the case of coal firing) are offset by increasing cylinder efficiency.
* Note: Wardale’s estimate for TGS drawbar efficiency differs significantly from the figure of 25% that he quotes as being Porta’s estimate for condensing third generation steam locomotives – see Second Generation Steam page of this website. However Wardale makes it clear (on page 501 of his book) that his figure applies to noncondensing locomotives and that “higher efficiency could only be obtained by expanding the steam to subatmospheric pressure and low temperature by means of condensing to counter the negative effect on the cycle efficiency of the restricted inlet steam temperature as done in stationary steam plant”.
Boiler efficiency can be defined as the amount of energy delivered from the boiler in the form of steam divided by the amount of energy delivered to the firebox in the form of fuel/chemical energy.
Boiler efficiency depends on the design on boiler and firebox, the type of fuel, and the draughting system. In the case of coalfired boilers, boiler efficiency declines linearly with rate of fuel feed, as discussed on the Grate Limit and Boiler Efficiency page.
Wall Effects and Condensation
Wall Effects: The term “wall effects”refers to the changes in steam temperature caused by temperature differentials between the steam and the walls of the cylinder, its end covers and the steam passages connecting to it.
When high temperature, high pressure steam enters the cylinder, it comes into contact with the relatively cool walls. The wall surfaces are cooler than the steam partly because they lose heat by conduction and radiation, but more importantly they are cooled by the steam itself as it expands during the release and exhaust phases of the power cycle. Thus it may be deduced the temperature of these surfaces varies around an average that is a little lower than the average steam temperature over the cycle. As a resuheatlt, not only are these surfaces cooler than the incoming steam and so absorb heat from it, but they are warmer than the outgoing steam and so give up heat to it.
The fact that heat is transferred from the hot to the cylinder walls during the expansion phase and heat is transferred to the walls during the beginning of the compression phase, means that both these phases are not adiabatic. In consequence, the expansion index (n in the equation PV^{n} = k) during these phases varies from a maximum (adiabatic) value of 1.3 to maybe 1.2 or in the case of short cutoff working and slow rotational speeds, even as low as 1.1.
Condensation: Because steam locomotives have traditionally been inadequately superheated, the term “wall effects” is often used synomymously with “condensation”.
Condensation of steam inside the cylinder results in a massive waste of energy as described in detail in Porta’s “compounding” paper published in Camden’s book “Advanced Steam Locomotive Development – Three Technical Papers“.
Porta describes wall effects as follows:
“Wall effect phenomena occur as follows. The cylinder cover, the steam passages and the piston (where the wall effects are most intense) are at a temperature between that of the live steam and the exhaust steam. Therefore, when the superheated steam makes contact with them, an energy drop takes place which show as a temperature drop. …. If the temperature of the walls is higher than the saturation temperature, no condensation occurs – the heat transfer being governed by “gas” laws, and is very small. But as shown by experimental data, if the temperature of the confining walls is below the saturation point at steamchest pressure, then condensation occurs even if the steam is highly superheated. This situation is the most frequent one in the life of steam locomotives: one of the causes is poor insulation, leading to heavy cooling down because of the intermittent nature of railway work.
The Author’s measurements, even if rough, show that when the cylinders are well warned up after a lengthy, strenuous pull, in an ordinary locomotive in which the steam temperature attains 400°C, the walls reach a temperature higher than 210°C only at cutoffs greater than ~20%. Therefore, at shorter cutoffs, condensation occurs and the machine works with wall effects approaching those of a saturated one.
The temperature drop (in the case of no condensation, later to be defined) is roughly inversely proportional to the cutoff and to the rotational velocity to the power of 0.3 (ω^{0.3}). Thus, in the case of shunting locomotives, single expansion engines work with wall effects corresponding to saturated engines working at very low speeds, say ~1 rps, hence very high.
A large number of tests reported by GUTERMUTH show that condensation in saturated engines fed with steam at 8 to 12 bar, between points 6 and 2, Fig. 1, amounts to 40 to 50% of the steam admitted to the cylinder (and even more). And this refers to STATIONARY engines. What can be expected in the case of well “ventilated” [i.e. poorly insulated] locomotive cylinders?”
Fig 1 referred to in the text above is copied below with some corrections. It is also slightly simplified for clarity:
In this diagram, the variable steam inlet pressure between points 1 and 2 is replaced by a straight horizontal line G between 1^{1} to 2^{11} representing the mean pressure of the steam entering the cylinder. The location of this line is derived by equating the hatched areas above and below it.
If steam entering the cylinder comes into contact with surfaces that are below the saturation temperature for the steam, then condensation will occur onto those surfaces. Each cubic centimeter of condensate representing perhaps 1000 cc of useful steam (depending on its pressure). In any case it represents a contraction or loss of steam that is represented by the segment 2^{1} to 2^{11}. This in turn results in an irrecoverable* loss of work or energy as represented by the hatched area H.
[* Porta points out that the condensate is likely to vapourize during the compression phase, thus recovering some energy that supplements the draught created by the exhaust system. This is offers no compensation for the loss of energy that might otherwise have been used for moving the locomotive and its train.]
In his “Compounding” paper, Porta goes on to say that:
“the basic principal concerning wall effects is to have at all times the temperature of the walls above the saturation point. It is therefore essential to have the highest possible steam temperature, even beyond the theoretical optimum. This point was missed by Churchward and all the British designers who followed him, with the possible exception of Gresley. The Americans also missed it.
The Author coined the expression “exaggerated insulation” (“hyperexaggerated” for fireless locomotives) which implies the substitution of sealedforlife fibreglass type material in place of cancercausing, and less efficient, asbestos, and also the concept of insulating the whole cylinder front end, including the smokebox saddle, and its bottom, comprising the frame. Perhaps the most important point is that of keeping the cylinders hot between the working spells inherent to the intermittent nature of railway service. The “sealedforlife” concept is a critical part of this; Wardale reported that in South Africa some 70% of the locomotives had no insulation at all!
A further attack on the problem could be the application of ceramic coatings on the surfaces: around 1890 Thurston (a very clever man) proposed painting these with a mixture of graphite and linseed oil. The Author tried it, but for trivial reasons the experiment was not followed up.
Another means of reducing wall effects to negligible proportions is to use steam jackets, but not those adopted by Chapelon for the 160A1 which required very complex castings. A welded fabrication is much simpler. These should be fitted to the cylinder covers, and around the steam passages, but NOT to the cylinder barrel. Jackets can also be fitted to the cylinders of fireless engines operating with saturated steam. Another difference from Chapelon’s scheme is that the jacket condensation is not mixed up with the main steam flow, but reinjected to the hot water reservoir.
Porta makes the point that condensation is equivalent to steam loss which necessarily increases steam consumption, and that this this directly reduces a locomotive power output as illustrated by his oftrepeated equation:
Porta concludes that wall effects or condensation can be minimised by:
 Using the highest possible superheat temperatures;
 Providing the best possible insulation of all external hot surfaces such as cylinders, cylinder covers, valves, steam chests and even the smokebox saddle to which steamchests and cylinders are connected;
 Providing steam heating to cylinder covers (but not cylinders);
 Using the smallest possible wheel diameter to maintain a high revolution rate;
 Using the smallest possible cylinder diameter and long stroke.
He also points out that:
 Short cutoff working results in lower surface temperatures and increased likelihood of condensation. Better to operate at >20% cutoff, illustrating one of the advantages of “compound” working;
 Condensation is inevitable during warmup periods – usually around 20 minutes of operation on full power. Shunting locomotives are likely to suffer from large condensation losses.
 Fireless locomotives operating on saturated steam are also likely to insuffer from large condensation losses.
Several pages of this website include text and diagrams copied from Porta’s “compounding” paper, including the pages covering steam leakage, clearance volume, incomplete expansion and triangular losses. 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.
Triangular Losses in Cylinders
Page Under Development
This page is still “under development”. Please contact Chris Newman at webmaster@advancedsteam.org if you would like to help by contributing text to this or any other page.
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 cutoff;
 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 precompression) 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, precompression 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 preadmission 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.
Conclusion:
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 longtravel 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.
Tractive Effort and Power
Tractive Effort
“Tractive effort” (TE) is the force applied by a locomotive for moving itself and a train. Tractive effort or tractive force is measured in kiloNewtons (kN) or pounds force (lbf) where 1 kN = 228.4 lbf.
As with “power” there are different methods of measuring tractive effort:
 Drawbar tractive effort – the force applied by a locomotive to the connection to its train. If the locomotive is running light (with no train) then its drawbar TE = 0.
 Wheelrim tractive effort – the force applied by a locomotive to the rails through its driving wheels. The difference between Wheelrim TE and Drawbar TE is the force required to move the locomotive in overcoming internal (mechanical), rolling and wind resistances.
 Indicated or Cylinder tractive effort – a hypothetical force estimated by adding to the Wheelrim TE the force required to overcome the frictional resistance of piston against cylinder, pistonrod against gland, crosshead against slidebar and the rotational resistance of big and smallend bearings. In the case of the 5AT (FDC 1.1 lines 46 and 47), the Wheelrim TE is estimated to be 93% of Cylinder TE at starting and 96% when running.
 Starting tractive effort – the pulling force exerted by a locomotive when starting from rest.
The commonly used formula for calculating a locomotive’s starting Tractive Effort is
The formula takes no account of piston rod diameter, which reduces the effective area of (and therefore the force applied to) the rear side of the piston. In the case of the 5AT (and a few other locomotive types) the presence of a tailrod reduces the effective area of both sides of the piston.
A locomotive’s starting tractive effort provides only an indication of the size of train that it can start. It does not measure the ability of the locomotive to pull a train at speed. This is because tractive effort reduces as speed increases. A locomotive that can maintain a high tractive effort at speed is a more “powerful” locomotive than one that cannot since Power = Tractive Force x Speed.
The relationship between TE and Speed for a variety of locomotives is illustrated in the diagram below (copied from page 499 of Wardale’s book “Red Devil and Other Tales from the Age of Steam“) in which it can be seen that the TE of the “Super Class 5 460 (5AT) remains higher than even the most powerful British Pacifics once their speed exceeds 70 km/h. The 5AT’s ability to maintain high TE at speed is a measure of its ability to deliver and make use of steam that is supplied to the cylinders i.e. “good breathing”. (The diagram can be compared to the Power – Speed diagram copied from the same page, which is shown on the Drawbar Power page of this section of the website.)
A locomotive’s tractive effort (at all speeds) is limited by its adhesive weight and the available coefficient of adhesion between wheel and rail, as discussed on the Adhesion page of this website.
The maximum speed that a locomotive can attain with any given train occurs when the locomotive’s drawbar tractive effort exactly equals the rolling resistance of the train (see the Rolling Resistance page of this website).
The acceleration that a locomotive can achieve with any given train can be calculated by applying Newton’s Second Law of Motion – i.e. by subtracting the rolling resistance of the locomotive and train from the locomotive’s wheelrim tractive effort, and dividing the difference by the total mass of the locomotive and train.
Locomotive Power
Power is defined as “the rate of doing work”. Common units of power in the metric system are Watts (W), kilowatts (kW), Megawatts (MW) and Gigawatts (GW), where 1 Watt = 1 Joule per second = 1 Newtonmetre per second. Alternative units of measurement are calories per second and kilocalories per hour (1 kW = 860 kcal/hr) Common imperial units of power are: Btu per second (1 Btu/s = 1.06 kW) and the Horsepower where 1 (British) hp = 0.746 kW.
Power can also be defined as the multiple of force and speed, from which it can be deduced that a locomotive’s power and tractive effort (TE) are intrinsically related: Power = TE x speed.
Livio Dante Porta liked to define a locomotive’s power in the following thermodynamic terms, as can be found in his papers on “Fundamental Principles of Steam Locomotive Modernization and Their Application to Museum and Tourist Railways” and “Fundamentals of the Porta Compound System for Steam Locomotives“:
In reference to the first equation, Porta writes:
“Thus the power is limited by [the amount of steam supplied by] the boiler, while the function of the cylinders is to extract the maximum work from the steam supplied”,
“The second equation shows that the [power] limit is determined by the ability of the boiler to burn as much fuel per hour as possible, but the resulting power is determined by the thermal efficiency.”
A common point of confusion in locomotive terminology is the difference between indicated power and drawbar power, the basic difference being that “indicated power” is the power developed in the cylinders, whereas “drawbar power” is the power delivered at the drawbar. These terms are defined on separate pages.
The word “indicated” comes from the use of “Indicator Diagrams” that before the electronic era were mechanically plotted to indicate the variation in steam pressure inside a cylinder against the piston position as it sweeps through the piston over the length of its stroke. Separate diagrams for both ends of the cylinder were usually being plotted on the same sheet of paper.
The cylinder’s power output is calculated from the area contained within the plotted curve. Thus it can be deduced that a locomotive’s power output can be increased by increasing the area contained inside the Indicator Diagram.
An example of a typical Indicator Diagram (single ended) is shown below which also points out the four phases of the cylinder cycle:

Admission which occurs from the moment that the steam inlet port opens (near the beginning of the piston stroke) until the moment of “cut off” when it closes. At the beginning of the piston stroke, the cylinder pressure is (or should be) the same as the steam chest pressure. As the piston moves, some reduction of pressure may occur if the steam chest and/or the ports are too small.

Expansion which occurs from the moment of “cut off” until the exhaust port opens (near the end of the piston stroke). During this time, the steam in the cylinder expands adiabatically (meaning no heat input) resulting in the reduction of its pressure. The relationship between pressure and swept volume can usually be estimated using the equation PV^{n}=K, where P is the steam pressure; V is the cylinder volume (including clearance volume); n is a constant, normally assumed to be 1.3 and K is a constant.

Exhaust which occurs on the return stroke between the moment the exhaust port opens until it closes again. During the exhaust phase, the cylinder pressure – or “back pressure” – is relatively constant, being governed by the steam flow through the exhaust ports and blast pipe. It can be readily seen that the area within the diagram can be dramatically increased by reducing the back pressure during the exhaust phase. This is the reason why simple modifications such as the fitting of double chimneys or Kylpor exhaust systems, result in immediate and dramatic improvement in locomotive performance.

Compression which occurs between the moment that the exhaust port closes until the inlet port opens. It is clear that the area within the Indicator Diagram can be maximised by making the exhaust phase as long as possible and the compression phase as short as possible. Some compression may nevertheless be desirable, as in the case of the 5AT where it provides “cushioning” for the pistons, counteracting the inertial forces that occur at the ends of each stroke and which would otherwise cause crankpin overstress at very high speeds.
The shape of the indicator diagram and losses that are represented by it, are discussed on several separate pages including:
Drawbar Power
The power output at the drawbar of a locomotive, the drawbar being the coupling between the locomotive and the train that it is hauling.
Drawbar power used to be measured by attaching a “dynamometer car” between the locomotive and its train. A dynamometer car incorporates a number of measuring devices including a calibrated spring for measuring the tractive force from the locomotive and a odometer wheel for accurately measurement of distance covered, and a timing device from which speeds can be calculated. In addition, the dynamometer car would house mechanical plotting devices and a team of people to monitor them. Nowadays an electronic loadcell can be fitted between the locomotive and its train and GPS used for measuring speed and distance, with data being logged and power outputs calculated on a laptop computer.
Drawbar power is the Indicated Power minus the mechanical losses in the locomotive’s motion and the rolling losses (including the wind losses) of the locomotive and its tender (see Resistance page). In the case of the 5AT, its drawbar power is diminished by the fitting of a large (80 tonne) tender. Because the rolling losses (and especially the wind losses) increase with speed, a locomotive’s drawbar power tends to peak at a lower speed than the Indicated Power.
Equivalent Drawbar Power
Equivalent drawbar power = drawbar power at constant speed on level tangent track. It eliminates the factors of acceleration and gradient/curvature resistance on the locomotive itself, the power to overcome which would be available at the drawbar at constant speed on level tangent track.
“Equivalent drawbar power vs. speed curves for various locomotives are shown below, copied from page 499 of David Wardale’s “The Red Devil and Other Tales from the Age of Steam” with the Tractive Effort curves removed for clarity.
[Note: The same diagram showing Tractive Effort vs. Speed can be viewed on the Tractive Effort page of this website.][It is interesting to note that the Standard “Britannia” Class 7 produced a slightly higher power output than the Rebuilt “Merchant Navy” Pacific.]
PowertoWeight Ratio
The PowertoWeight ratio of a car is a measure of its ability to accelerate. A steam locomotive’s ability to accelerate is governed by its the ratio of its “power : total train weight” ratio and by its adhesive weight and adhesion coefficient (ignoring resistance factors).
The PowertoWeight ratio of a steam locomotives is nevertheless an important meaure, however its implications are more nuanced than for a car. David Wardale explains the importance of having a high PowertoWeight ratio on page 277 of his book as follows:
…. for high speed operation, a high Power : Weight ratio is essential. It implies the need for a small boiler, requiring highly efficient draughting, and high combustion rates, requiring an efficient combustion system. Failure to realize this means that at high speed most of a locomotive’s power is absorbed in pulling the locomotive itself. In fact any locomotive has a ‘zero drawbar power and thermal efficiency speed’ at which all its power is used to pull itself along, this speed being largely a function of its inbuilt power : weight ratio. That this ratio was not high enough in steam locomotives was the basic reason why steam was perceived as being unsuitable for the accelerated services which many railway administrations, especially in Europe, saw as essential if rail transport was to remain competitive. It was, however, only an inherent characteristic of most First Generation Steam (FGS) locomotives, not of steam traction per se.
On page 273 of his book he compares the PowertoWeight ratio of the Red Devil with other FGS locomotives as follows:
The Drawbar PowertoWeight ratio [of No 3450, the Red Devil] based on the engine weight only (i.e. excluding the tender) was 23.0 kW/ton calculated from the maximum recorded sustained equivalent drawbar power at 74 km/h, and 24.4 kW/ton based on the predicted maximum drawbar power at 100 km/h.
The maximum Drawbar PowertoWeight ratio in kW per ton of engine weight for some other high power coalfired locomotives were as follows:
 British Railways ‘Coronation’ class 462: 17.1 (best British figure based on transitory power)’
 German State Railways 45 class 2102: 17.5 (approximate)^{1}
 French National Railways 240P class 480: 23.3
 New York Central `Niagara’ class 484: 18.8 (representative of the very best American practice)
 Rio Turbio Railway 2102: 20.6 (at 50 km/h, the maximum line speed)
 Porta’s experimental 480: 23.2
In respect of power capacity relative to size 3450 was therefore up to the best standards hitherto achieved despite it being a 2cylinder simple expansion locomotive with moderate boiler pressure burning mediocre quality coal, this last factor being of great significance as very high power output from steam locomotives generally depended on burning high grade coal. Yet noone should imagine that it represented a performance ceiling for the classical Stephensonian steam locomotive. Porta’s Second Generation Steam 2100 proposal was designed to give a rated Drawbar PowertoWeight ratio of 32.5 kW/ton, and even with simple expansion 29 kW/ton should have been possible for a mediumspeed machine if starting the design with a clean sheet of paper.
The predicted Drawbar PowertoWeight ratio for the 5AT compared well with the above figures. With a maximum sustainable drawbar power at constant speed on level tangent track (and trailing a high capacity tender) of 1890 kW and an engine weight of 80 tons, its Drawbar PowertoWeight ratio would have been 23.6 kW/ton.
[Note: Wardale uses the word “ton” to mean “tonne” in SI units.]
^{ }
Steam Terms
Equivalent Evaporation
Equivalent evaporation = evaporation from and at 100°C. Evaporation figures thus expressed eliminate the effects of different feedwater and superheat temperatures, and are therefore a true measure of comparison between different boilers.
[Extract from letter from Dave Wardale to Chris Newman, 5th April 2001.]
Specific Steam Consumption
Specific Steam Consumption is defined as the steam consumed by a locomotive’s cylinders per unit output of power. It is typically measured in kg/kWh or kg/KJ.
A locomotive’s Specific Steam Consumption carries important implications as may be deduced from one of Porta’s favourite equations:
Thus for any given boiler output, a locomotive’s power can be increased by reducing its specific steam consumption – in particular, by increasing its cylinder efficiency and reducing steam leakage. Or as Porta put it, “the power is limited by [the amount of steam supplied by] the boiler, while the function of the cylinders is to extract the maximum work from the steam supplied”.
In Section 4 of his “Compounding” paper, Porta makes the observation:
“In steam locomotives, one should note that all the losses, except for incomplete expansion, are approximately constant for a given rotational speed. Hence, the aim is to have a longer cutoff but, given that this steeply increases the incomplete expansion losses, a compromise results at 20% to 30% (15% to 20% for the author’s proposals). Thus, the claims for poppet valves concerning their ability to work with very short cutoffs are illusory as they do not lead to low specific steam consumption because of these constant losses.
But there are economic reasons too. The Americans, who have the perverse habit of hooking as many cars as possible to their locomotives, force them to work at long cutoffs to get as high an α coefficient as possible so as to have a good use of the (expensive) adhesion weight. This of course leads to a high specific steam consumption, hence the need for massive evaporation, hence a massive boiler, hence idle axles to support a huge firebox, hence a gigantic tender, hence plants to supply coal enroute, hence immense coal stocks, hence diesel locomotives with a higher thermal efficiency (under test conditions) even if they cost twice as much and justify the Gulf War to supply them with oil.”
In the same paper, Porta also refers to Specific Steam Consumption in relation to the TESpeed diagrams below, which appear under the heading “Boiler Size”. He introduces the diagrams as follows:
“The operating variables of any locomotive working with the throttle full open can be defined, for a fully warmed up condition, by (any) two of them. For example: tractive effort vs. speed; steam production vs. speed; cut off vs. speed, etc. In Fig. 32A, for example, the constant cutoff lines have been plotted on a TE vs. Speed diagram. There is a line corresponding to the maximum cutoff, and various lines for the various running cutoffs. As a first approximation, they are straight lines whose inclination is greater, the greater the imperfection of the internal streamlining. In ordinary locomotives in which the internal streamlining is poor, the lines have an envelope: no combination of speed and cutoffs make it possible to invade the zone M (Fig. 32B).
In Fig. 32A, the lines corresponding to constant evaporation have been drawn (lines 3) and also the lines for constant specific steam consumption (lines 5). They show a zone (hatched) in which this consumption is minimal (zone 6) and also a zone (zone 7 crosshatched) in which it decreases (very much in the case of single expansion engines) due to the increase of the incomplete expansion losses. There is also a zone in which the various constant losses (leakage, wall effects) increase specific steam consumption – this is important in the case of shunting engines. Obviously, the aim of the designer is to provide a maximum area covered with consumptions differing as little as possible from the optimum.”
Figs. 32A and 32B: Characteristic Lines
Notes on Fig 32: In Fig. 32A, Straight lines (1) are constant cutoff lines, (2) being the one corresponding to full gear. The various hyperbolalike lines (3) correspond to constant evaporation. Selecting one of them, such as (4) allows the provision of a definite boiler size. The hatched area (R) corresponding to the overload concept.
Curves (5) refer to constant specific consumption, the hatched area (6) indicating the combination of speedtractive effort in which the consumption is at a minimum. Area (7) refers to low speed, low tractive effort characteristic of shunting services.
So far, the above refers to engines designed with good internal streamlining (a RARE case indeed!). Fig 32B is the common case in which the cutoff lines are so much inclined that they have an envelope (8): this corresponds to the American concept of “capacity power”; no combination of speed and tractive effort allows getting into the M region. Within the envelope area, the specific steam consumption is very high: this explains the huge size of American boilers and tenders.
Wardale also refers to Drawbar Specific Steam Consumption in his book, defining it (on page 273) as:
He goes on to point out that:
“Drawbar Specific Steam Consumption is therefore influenced by the power required to move the locomotive and as the measured values of this parameter were thought to be too high, the drawbar Specific Steam Consumption data [for the Red Devil] was distorted, especially at higher especially at higher speeds and lower steaming rates. (From this equation, it can be readily seen that the drawbar Specific Steam Consumption of a locomotive which was not capable of generating high power relative to its weight, was bound to suffer at high speed, however good the indicated Specific Steam Consumption was – e.g. Duke of Gloucester.”
In the Fundamental Design Calculations for the 5AT (see FDC 1.3), Wardale gives figures of minimum indicated Specific Steam Consumption for the Duke of Gloucester as 12.2 lb/hph and for the SNCF 141P Class 4cyl. compound 282 as 11.2 lb/hphr, as compared to 11.24 lb/hph (= 5.1 kg/hphr or 1.9 kg/MJ) for the 5AT.
The FrontEnd Limit
As discussed on the Grate Limit page, the grate limit occurs when any increase in the rate of fuel delivery produces no increase in evaporation. In other words it represents the maximum rate of heat emission that a firebox can deliver beyond which point any additional fuel added to the firebox produces no additional steam.
The Front End Limit is a draughting limitation. In his paper titled “Two Point Four Pounds per Ton and The Railway Revolution“, Doug Landau defines the Front End Limit as occurring when the available excess air falls below about 20% and complete combustion can no longer be achieved. If this occurs prematurely, the locomotive concerned would be deemed a ‘poor steamer’. It could also be set by the designer at a value that would provide adequate steam, while at the same time avoiding ‘uneconomic’ combustion rates. The BR Standard locomotives were designed on this basis.
It may be observed from the above that the Front End Limit occurs at a lower steaming rate than the grate limit.
In the same paper, Doug goes on to describe what he calls the “Discharge Limit” which he differentiates from the Front Edn Limit as follows:
Discharge Limit: This is also sometimes described as the ‘Front End Limit’, but it is quite different to the condition described above. It occurs when the steam exhaust velocity reaches the speed of sound. At this point theory has it that the pressure/draught relationship breaks down. Curiously however, there are recorded instances of this limit being exceeded without apparent distress. It does however involve very high back pressures upwards of 14 lbs/sq.in., and was definitely something best avoided.
Dave Wardale offers the opinion that the first of the above definitions is correct, but adds that “the cause of excess air falling to a useable limit is due to the blast pipe and chimney characteristic. Although BR claimed to have designed for this, Porta pointed out that whether the BR front end limit was by design or because they couldn’t do any better was an open question.”
The above graph of Boiler Efficiency vs. Specific Firing Rate includes a curve marked “Equivalent Evaporation”. Equivalent Evaporation is used as a means of comparing the performance of boilers under a standard set of conditions, and it might be most simply defined as “the quantity of water at 100°C that a boiler can convert into dry/saturated steam at 100°C from each kJ of energy that is applied to it. This defines it in terms of kg (of water/steam) per kJ of energy. However it is sometimes defined in units of kg water per kg of fuel, and (as in the case of the graph above) kg water/steam per hour.
A more exact definition of Equivalent Evaporation in units of kg/hr comes from “Thermal Engineering” pages 608/9 by R.K. Rajput (see Google Books):
“Generally the output or evaporative capacity of the boiler is given as kg of water evaporated per hour but as different boilers generate steam at different pressures and temperatures (from feed water at different temperatures) and as such have different amounts of heat ; the number of kg of water evaporated per hour in no way provides the exact means for comparison of the performance of the boilers. Hence to compare the evaporative capacity or performance of different boilers working under different conditions it becomes imperative to provide a common base so that water be supposed to be evaporated under standard conditions. The standard conditions adopted are: Temperature of feed water 100°C and converted into dry and saturated steam at 100°C. As per these standard conditions 1 kg of water at 100°C necessitates 2257 kJ (539 kcal in MKS units) to get converted to steam at 100°C.
“Thus the equivalent evaporation may be defined as: the amount of water evaporated from water at 100°C to dry and saturated steam at 100°C.
“Consider a boiler generating m_{a} kg of steam per hour at a pressure p and temperature T.
Let h = Enthalpy of steam per kg under the generating conditions.

 h = h_{f} + h_{fg} ……. Dry saturated steam at pressure p
 h = h_{f} + x_{hfg} ……. Wet steam with dryness fraction x at pressure p
 h = h_{f} + h_{fg} + c_{p} (T_{sup} – T_{s}) ….. Superheated steam at pressure p and temperature T_{sup}
 h_{f1} = Specific enthalpy of water at a given feed temperature.
Then heat gained by the steam from the boiler per unit time = m_{a} x (h – h_{f1})
The equivalent evaporation (m_{e}) from the definition is obtained as:
The evaporation rate of the boiler is also sometimes given in terms of kg of steam /kg of fuel. The presently accepted standard of expressing the capacity of a boiler is in terms of the total heat added per hour.
An alternative definition is offered by Applied Thermodynamics by Onkar Singh as follows:
“For comparing the capacity of boilers working at different pressures, temperatures, different final steam conditions etc, a parameter called “equivalent evaporation” can be used. Equivalent evaporation actually indicates the amount of heat added in the boiler for steam generation. Equivalent evaporation refers to the quantity of dry saturated steam generated per unit of time from feedwater at 100°C to steam at 100°C at the saturation pressure corresponding to 100°C. Sometimes it is called equivalent evaporation from and at 100°C. Thus mathematically it could be given as:
For a boiler generating steam at ‘m’ kg/h at some pressure ‘p’ and temperature ‘T’, the heat supplied for steam generation = m x (h – h_{w}), where h is the enthalpy of final steam generated and h_{w} is enthalpy of feedwater. Enthalpy of final steam shall be:

 h = h_{f} + h_{fg} = h_{g} for final steam being dry saturated steam (h_{f}, h_{fg} and h_{g }are used for their usual meanings),
 h = h_{f} + x_{ .} h_{fg} for wet steam as final steam,
 h = h_{g} + c_{p sup.steam} . (T_{sup} – T_{sat}) for superheated final steam_{.}
Equivalent evaporation (kg/kg of fuel) =
Equivalent evaporation is thus a parameter which could be used for comparing the capacities of different boilers.”
Note – the last equation purports to express equivalent evaporation in units of kg/kg of fuel, but in fact the units are actually in kg/hr.]
Drafting & Combustion
The Grate Limit as it relates to Boiler Efficiency
On page 78 of his book The Red Devil and Other Tales from the Age of Steam, Dave Wardale defines the Grate Limit for a (normal) locomotive firebox as follows:
The grate limit is the point “at which even by firing more coal and supplying more combustion air, no more steam could be produced.”
Put another way, it is the point at which the rate of firing fuel into the firebox exactly equals the rate at which unburned fuel is carried out of the firebox by entrainment in the combustion air.
Wardale quotes an equation derived by L.H. Fry in 1924 that (effectively) relates the grate limit to boiler efficiency as follows:
E_{b} = k_{1} – k_{2} x M/G, where
 E_{b} = Boiler Efficiency
 M = Firing Rate
 G = Grate Area
 k_{1} = predicted boiler efficiency at zero firing rate
 k_{2} = the slope of the graph relating boiler efficiency to firing rate.
The equation is illustrated in graphical form below (Fig 20 in Wardale’s book):
The above equation is empirical, yet it is one that produces a fascinating insight – namely that the boiler efficiency at the grate limit is exactly 50% of the predicted efficiency at zero firing rate.
In the diagram, k_{1} is the maximum predicted boiler efficiency (at zero firing rate) and k_{2} is the slope of the straight line relating efficiency to firing rate. [A simple mathematical proof that (based on Fry’s equation) the boiler efficiency at the grate limit is exactly half the maximum efficiency is given further below. A definition of Equivalent Evaporation is also provided on a sepate page.]
Wardale goes on to demonstrate that Fry’s equation represents reality, being demonstrated in a boiler test conducted on a Pennsylvania Railroad M1a 482 locomotive. Here Wardales adjusts his definition of the grate limit as follows:
The grate limit was the point at which “the heat liberation rate in the firebox was a maximum, which for all practical purposes occurred when the fuel entrained in the draught and escaping unburnt equalled the amount of fuel actually burned, this point being linked to the start of gross firebed fluidisation.”
In other words, the grate limit is reached when half the fuel that is fired into the firebox escapes from the chimney. He illustrates this with the diagram below taken from Fig 21 on page 80 of his book, to which the percentage figures on the right have been added [including an approximate division of “evaporation” into latent and sensible heat]:
It should be noted that Fry’s equation does not hold for GPCS fireboxes. Indeed, the fact that it does not hold is one of the great advantages that GPCS fireboxes offer.
A simple mathematical proof that, based on Fry’s equation, boiler efficiency at the grate limit is exactly half the maximum efficiency, is as follows:
Fry’s Equation: E_{b} = k_{1} – k_{2} x M/G, where:
 E_{b} = Boiler Efficiency
 M = Firing Rate
 G = Grate Area
 k_{1} = predicted boiler efficiency at zero firing rate
 k_{2} = the slope of the graph relating boiler efficiency to firing rate.
Boiler efficiency may also be defined as the amount of energy released from the boiler in the form of steam divided by the amount of energy released in the firebox from the fuel.
Thus if the Steaming Rate = S, then E_{b} = S ÷ M/G
Thus M/G = S/E_{b}
Substituting this in Fry’s equation we get: E_{b} = k_{1} – k_{2} x S/E_{b}
from which: E_{b}^{2} = k_{1}.E_{b} – k_{2}.S and thus: S = k_{1}.E_{b}/k_{2} – E_{b}^{2}/k_{2}
From calculus, we know that S reaches a maximum (or minimum) when the slope of the curve = zero. This occurs when
i.e. when …..
Thus the maximum steaming rate occurs at the point where the boiler efficiency is half the predicted value at zero firing rate.
The FrontEnd Limit
As discussed on the Grate Limit page, the grate limit occurs when any increase in the rate of fuel delivery produces no increase in evaporation. In other words it represents the maximum rate of heat emission that a firebox can deliver beyond which point any additional fuel added to the firebox produces no additional steam.
The Front End Limit is a draughting limitation. In his paper titled “Two Point Four Pounds per Ton and The Railway Revolution“, Doug Landau defines the Front End Limit as occurring when the available excess air falls below about 20% and complete combustion can no longer be achieved. If this occurs prematurely, the locomotive concerned would be deemed a ‘poor steamer’. It could also be set by the designer at a value that would provide adequate steam, while at the same time avoiding ‘uneconomic’ combustion rates. The BR Standard locomotives were designed on this basis.
It may be observed from the above that the Front End Limit occurs at a lower steaming rate than the grate limit.
In the same paper, Doug goes on to describe what he calls the “Discharge Limit” which he differentiates from the Front Edn Limit as follows:
Discharge Limit: This is also sometimes described as the ‘Front End Limit’, but it is quite different to the condition described above. It occurs when the steam exhaust velocity reaches the speed of sound. At this point theory has it that the pressure/draught relationship breaks down. Curiously however, there are recorded instances of this limit being exceeded without apparent distress. It does however involve very high back pressures upwards of 14 lbs/sq.in., and was definitely something best avoided.
Dave Wardale offers the opinion that the first of the above definitions is correct, but adds that “the cause of excess air falling to a useable limit is due to the blast pipe and chimney characteristic. Although BR claimed to have designed for this, Porta pointed out that whether the BR front end limit was by design or because they couldn’t do any better was an open question.”
Mechanical Terms
Tractive Effort
“Tractive effort” (TE) is the force applied by a locomotive for moving itself and a train. Tractive effort or tractive force is measured in kiloNewtons (kN) or pounds force (lbf) where 1 kN = 228.4 lbf.
As with “power” there are different methods of measuring tractive effort:
 Drawbar tractive effort – the force applied by a locomotive to the connection to its train. If the locomotive is running light (with no train) then its drawbar TE = 0.
 Wheelrim tractive effort – the force applied by a locomotive to the rails through its driving wheels. The difference between Wheelrim TE and Drawbar TE is the force required to move the locomotive in overcoming internal (mechanical), rolling and wind resistances.
 Indicated or Cylinder tractive effort – a hypothetical force estimated by adding to the Wheelrim TE the force required to overcome the frictional resistance of piston against cylinder, pistonrod against gland, crosshead against slidebar and the rotational resistance of big and smallend bearings. In the case of the 5AT (FDC 1.1 lines 46 and 47), the Wheelrim TE is estimated to be 93% of Cylinder TE at starting and 96% when running.
 Starting tractive effort – the pulling force exerted by a locomotive when starting from rest.
The commonly used formula for calculating a locomotive’s starting Tractive Effort is
The formula takes no account of piston rod diameter, which reduces the effective area of (and therefore the force applied to) the rear side of the piston. In the case of the 5AT (and a few other locomotive types) the presence of a tailrod reduces the effective area of both sides of the piston.
A locomotive’s starting tractive effort provides only an indication of the size of train that it can start. It does not measure the ability of the locomotive to pull a train at speed. This is because tractive effort reduces as speed increases. A locomotive that can maintain a high tractive effort at speed is a more “powerful” locomotive than one that cannot since Power = Tractive Force x Speed.
The relationship between TE and Speed for a variety of locomotives is illustrated in the diagram below (copied from page 499 of Wardale’s book “Red Devil and Other Tales from the Age of Steam“) in which it can be seen that the TE of the “Super Class 5 460 (5AT) remains higher than even the most powerful British Pacifics once their speed exceeds 70 km/h. The 5AT’s ability to maintain high TE at speed is a measure of its ability to deliver and make use of steam that is supplied to the cylinders i.e. “good breathing”. (The diagram can be compared to the Power – Speed diagram copied from the same page, which is shown on the Drawbar Power page of this section of the website.)
A locomotive’s tractive effort (at all speeds) is limited by its adhesive weight and the available coefficient of adhesion between wheel and rail, as discussed on the Adhesion page of this website.
The maximum speed that a locomotive can attain with any given train occurs when the locomotive’s drawbar tractive effort exactly equals the rolling resistance of the train (see the Rolling Resistance page of this website).
The acceleration that a locomotive can achieve with any given train can be calculated by applying Newton’s Second Law of Motion – i.e. by subtracting the rolling resistance of the locomotive and train from the locomotive’s wheelrim tractive effort, and dividing the difference by the total mass of the locomotive and train.
Adhesion and Adhesive Weight
Adhesion is the frictional resistance that prevents a locomotive’s driving wheels from slipping on the rail. Available adhesion depends on the conditions of both the rail and the wheel. With a clean dry wheel running on a clean dry rail the “frictional coefficient” between them may be as high as 0.35 or 35%. The presence of oil and other contaminents may reduce it to near zero.
Adhesive Weight is that part of a locomotive’s weight that is supported by its driving wheels. In the case of modern BoBo and CoCo diesels and electrics, 100% of their weight is supported by their driving (or driven) wheels which allows them to deliver a very large starting Tractive Effort. The same applies to 060T steam engine, however with 060 tender engines, perhaps no more than 70% of its total weight may be supported by its drivers. The greater the number of carrying wheels, the less weight is available for adhesion, and in the case of a typical Pacific locomotive less than 50% of total weight may be available for adhesion. This has serious consequences in terms of limiting the locomotive’s wheelrim tractive effort and its drawbar power.
Coefficient of Friction is defined as the frictional resistance between two bodies and the “normal” force applied between them. In the case of a locomotive, the Frictional Coefficient = available frictional resistance ÷ adhesive weight on the driving wheels.
Adhesion Factor is the inverse of Friction Coefficient, i.e. Adhesion Factor = adhesive weight on the driving wheels ÷ available frictional resistance.
Frictional Coefficient between Wheels and Rail: A clean dry wheel on a clean dry rail may have a coefficient of friction as high as 0.35 or 35% at zero speed. This can drop to 0.25 in wet conditions, and very much lower if the rail is contaminated with lubricating substances such as ice, oil and leaves.
However the frictional coefficient between wheel and rail is not constant: due to “elastic slip” it falls as speed rises which is why wheel slip can (and does) occur at speed. Curves showing the relationship between speed and friction coefficient are shown below (taken from Koffman’s equations):
Estimating a locomotive’s adhesion limit is further complicated by the fact that its wheelrim tractive effort varies considerably over each wheel revolution, even at high speed. As a result, momentary slipping can occur where the tractive force peaks and this can, in unfavourable circumstances, initiate fullscale slip at low and high speeds.
For further information, see the Advanced Adhesion page in the Principles of Modern Steam section of this website.
Locomotive and Train Resistance
A locomotive’s tractive force is required to overcome the resistance to motion of both locomotive and train. When the tractive force is greater than the resistance, then the train will accelerate in accordance with Newton’s law of motion: Force = Mass x Acceleration or Acceleration = Force ÷ Mass. If the tractive force is equal to the resistance, then the train will travel at constant speed. If the tractive force is less than the resistance, then the train will slow down.
There are various forms of resistance that need to be considered.
Locomotive Resistance

Starting resistance, associated with static friction which is usually higher than dynamic friction – i.e. it needs to be overcome before the locomotive will start moving;

Internal (mechanical/frictional) resistance from the locomotive’s motion etc;

Rolling resistance deriving from axle bearings and wheels on rail – it increases in proportion to the speed;

Wind resistance increases in proportion to the square of the speed;

Gravitational resistance when on an incline – equals the locomotive weight x the gradient;

Rail curvature resistance applies only to curved track – being proportional to the curvature of the track (or inversely proportional to the radius of curvature).
Train Resistance

Starting resistance, associated with static friction which is usually higher than dynamic friction – i.e. it needs to be overcome before the train will start moving. In the case of roller bearing stock, this value is close to zero, but can be significant where journal bearings are fitted to wagons;

Rolling resistance deriving from axle bearings and wheels on rail – it increases in proportion to the speed;

Wind resistance increases in proportion to the square of the speed;

Gravitational resistance when on an incline – equals the train weight x the gradient;

Rail curvature resistance – being proportional to the curvature of the track (or inversely proportional to the radius of curvature), but applying only to the length of train on the curved track.
In addition, inertial resistance has also to be considered (e.g. when a train is being accelerated).
The term “specific resistance” means the resisting force per tonne of (train) weight. It is pertinent to note that the specific resistance of empty freight trains and of passenger trains is much higher than that of loaded freight trains. This is largely because the wind resistance of an empty or lightly loaded train is the same as or higher (see below) than that of a full train. Since the wind resistance becomes dominant at higher speeds, the total resistance per tonne of train weight becomes higher. In the case of opentopped wagons, the wind resistance of empty wagons is higher than of full wagons because of the increased in air turbulence inside and around the empty wagon bodies. This is illustrated in the graphs at the bottom of this page.
It might also be mentioned that wind resistance is higher when the wind comes from an angle to the direction of the train than when the wind is directly headon to the train. This is because the wind has more impact on the ends of each wagon or carriage than when the wind is directly in front of the train where slipstreaming from one wagon or carriage to the next reduces drag.
Train resistance can be measured by the force in the coupling connecting the locomotive to its train (e.g. using a dynamometer car, or in latter days using an electronic load cell). Locomotive rolling resistance is harder to measure.
There are no exact methods for estimating locomotive or train resistance and reliance has to be placed on empirical formulae based on measured values. Almost every country’s railway has its own formulae for estimating rolling resistance, all (presumably) being based on recorded measurements.
Most resistance formulae are divided into three sections:

a fixed constant, representing a fixed part of the rolling resistance,

a term proportional to the train speed, representing a variable part of the rolling resistance, and

a term proportional to the square of the train speed representing wind resistance.
One of the more recent national train resistance formula to be developed was the Serbian one that was formulated in 2005 based on tests undertaken by M. Radosavljevic from the Department of Mechanical Engineering, the Institute of Transportation in Belgrade. He wrote a paper on his findings entitled “Measurement of Train Traction Characteristics” published in the Proceedings of the Institution of Mechanical Engineers Vol 220 Part F: J. Rail and Rapid Transit. A copy of the formulae and their graphical representation is copied below. (See below for further comparisons)
The above formulae are compared with additional train resistance formulae in the Excel graph below [click on the image for a fullsized PDF image.] The additional formulae covered are as follows

China National Railways (full wagons)

China National Railways (empty wagons)

Canadian National Railway (full wagons)

An (unnamed) Australian mineral railway (full wagons)

An (unnamed) Australian mineral railway (empty wagons)

Koffman’s formula for BR carriages as used by Wardale in FDC 1.1.
It will be readily seen that (as explained above) the specific resistance of empty wagons is much higher than that of full wagons and that for BR Carriages is also high by comparison with loaded wagons (due to their relatively light loaded weight). On the other hand the majority of formulae give very similar estimates for the specific rolling resistance of full wagons, with the exception of the (unnamed) Australian mineral railway figures which show much lower figures. This is probably because the wagons are much larger and more heavily loaded than those in the other countries listed.
Wardale’s Resistance Calculations
In FDC 1.1 (line 24) and FDC 1.2 (line 2) Wardale uses the following formulae for calculating locomotive and carriage rolling resistence (respectively):

Locomotive rolling resistance: R ≈ (45 + 0.24v + 0.0036v2) N/Tonne

Carriage rolling resistance: R = (1.1 + 0.021v + 0.000175v2) kg/tonne – from Koffman applying to BR coaches.
where R is the specific rolling resistance in N/Tonne of weight and v = speed in km/h.

In FDC 1.3 line 164, Wardale quotes Koffman to assume a specific starting resistance on level tangent track for roller bearing stock of 7.0 kg/tonne (= 6.99 N/kN).

In FDC 1.3 line 166, Wardale uses a figure of 100 N/tonne (=10.2 N/kN) for the specific starting resistance for a locomotive.
Chinese National Railways Resistance Calculations
For the record, various resistance formulae used by Chinese National Railways are summarized as follows, using the notation:

R = resistance in Newtons

W = weight in kN (not tonnes)

V = speed in km/h

Grade = slope in ^{o}/_{oo} [e.g. 1 in 100 = 10^{o}/_{oo}]

Rad = curved track radius in metres.

L_{t} = length of train in metres.

L_{c} = length of curve in metres; L_{c }= L_{t} where L_{t} ≤ L_{c}
Rolling Resistance of Locomotives – in Newtons:

QJ with 6 axle tender; R= W x [(1.09 + 0.0038 V + 0.000586 V^{2}) + Grade + 600/Rad]

QJ with 4 axle tender: R= W x [(0.70 + 0.0243 V + 0.000673 V2) + Grade + 600/Rad]

JS: R= W x [(0.74 + 0.0168 V + 0.000700 V2) + Grade + 600/Rad]

SY: R= W x [(0.74 + 0.0168 V + 0.000700 V2) + Grade + 600/Rad]
Rolling Resistance of Wagons – in Newtons:

Roller Bearing Wagons: R= W x [(0.92 + 0.0048 V + 0.000125 V2) + Grade + 600/Rad x L_{c}/L_{t}]

Journal Bearing Wagons: R= W x [(1.07 + 0.0011 V + 0.000236 V2) + Grade + 600/Rad x L_{c}/L_{t}]

Empty Wagons: R= W x [(2.23 + 0.0053 V + 0.000675 V2) + Grade + 600/Rad x L_{c}/L_{t}]
Starting Resistance

Locomotives: 8 N/kN (weight)

Roller bearing wagons: 3.5 N/kN

Journal bearing wagons: 5 N/kN
Canadian National Railway Resistance Formulae
Chapter 2.1 of AREMA (American Railway Engineering and MaintenenceofWay Association) Manaul for Railway Engineering, dated 1999 and titled “Resistance to Movement” provides data and formulae used by Canadian National Railways. The document can be downloaded from this website by clicking here.
Thermodynamics
Draft Text Only – Readers’ suggestions and inputs are welcome
Introduction
Oliver Bullied, disciple of Gresley and famously progressive CME of the Southern Railway, is often quoted as saying “Thermodynamics never sold a single locomotive” (or words to that effect) when commenting on Chapelon’s contemporary locomotive developments in France. Whether true or apocryphal, the remark exemplifies the lack of understanding of Thermodynamics that was widely prevalent within the locomotive engineering fraternity of his day. Indeed it remains poorly understood by many engineers today, and is a complete mystery to most laymen.
In fact, the study of Thermodynamics is still evolving to the extent that it now spreads far beyond the mysteries of steam power that inspired its early development. The author of the web page http://thermodynamicstudy.net/history.html offers a broad view of its modern day scope:
The history of thermodynamics [is] not only one of the most interesting but one of the most dramatic episodes to be found in the story of the intellectual progress of the human mind. Starting in an investigation of a purely practical problem of engineering economics, it has grown into a body of doctrine of profound philosophical significance, with consequences which permeate the thinking of men on many subjects, from those with the most practical use to the problems of cosmology.
This page and the subpages under it, attempt to explain the rather esoteric and abstract concepts that underlie the subject of thermodynamics as it applies to steam traction, using terms that it is hoped will be more readily understood than those found in most texts on the subject.
History of Thermodynamics:
Many scientists of past ages could claim to be the “father of Thermodynamics” but it is probably most useful to give the credit to Nicolas Léonard Sadi Canot, a young French military engineer who in 1824 set out to determine how the greatest amount of mechanical work could be obtained from a given amount of heat. In so doing, he invented the idea of the Carnot Cycle, being an idealized concept from which the maximum theoretical efficiency of any heat engine can be determined through the simple equation:
Carnot efficiency = (1 – T_{2}/T_{1}) where T_{1} and T_{2} are the temperatures
of the heat source and heat sink respectively, measured in ^{o}K.
It was not until the 1920s that André Chapelon began to apply the theories of Thermodynamics to the design of steam locomotives, with immediate and dramatic results. Unfortunately his work remained poorly understood in most steam locomotive design offices around the world and it was only in the 1950s that Livio Dante Porta took up the mantle and continued the work that Chapelon had started.
Important amongst the many lessons that Chapelon (and Porta) learned from Thermodynamics comes from Carnot’s simple equation which explains the importance of high temperature superheat since it shows that an engine’s efficiency is critically affected by its temperature. In the case of a typical “first generation” steam locomotive operating at a superheat temperature of (say) 350^{o}C and with an exhaust steam temperature of (say) 180^{o}C, its theortetical (maximum) Carnot efficiency would be 27%, whereas the 5AT operating at a superheat temperature of 450^{o}C and exhausting at 183^{o}C, its theortetical (maximum) Carnot efficiency is 37%.
[Note: Carnot’s theorem derives efficiency values that are purely theoretical. His equation does not apply specifically to steam engines but to all types of heat engine – steam, diesel, Stirling or any other. The purpose of comparing Thermodynamics dictates through Carnot’s Theorem that any heat engine’s efficiency is limited by the temperature differences between which it operates. See also Superheating page]
Laws of Thermodynamics
The study of thermodynamics is defined by three relatively simple laws:

First Law – energy can be neither created nor destroyed, or “energy is conserved”;

Second Law – energy will tend to dissipate from a hot or high energy body to a cold or low energy sink – or “heat cannot spontaneously flow from a cold body to a hot one”;

Third Law – defines absolute zero as equalling 273^{o}C, being the hypothetical point at which energy becomes zero.
In fact the Second Law can be written in many different ways. Here is another one:

Alternative Second Law: It is impossible to extract an amount of heat from a hot source and use it all to do work. Some amount of heat must be exhausted to a cold sink. This precludes a “perfect” heat engine.
Whilst the laws themselves are simple enough, the interpretation of them leads to complications. Most notably, the Second Law involves understanding the concept of Entropy. The concepts of Entropy and Enthalpy are described in separate subpages, together with a brief explanation of what “Steam Tables” are about.
There are a wide range of interpretations covering the concepts of Thermodynamics that can be found on several websites including the following:
 http://en.wikipedia.org/wiki/Carnot_heat_engine
 http://fr.wikipedia.org/wiki/Sadi_Carnot_(physicien)
Page Under Development
This page is still “under development”. Please contact the webmaster@advancedsteam.org if you would like to help by contributing text to this or any other page.
Definitions of thermodynamic concepts such as Entropy and Enthalpy are provided on separate pages of this website. Numerous useful (and often diverse) definitions of these and other terms can be found on the Internet.
A particularly useful (and brief) discussion can be found at http://www.mathpages.com/home/kmath184/kmath184.htm, the content of which is available from this website in PDF form (click here).
Adiabatic Process
An adiabatic process is one in which there is no transfer of heat between the system and its surroundings. Thus in thermodynamic nomenclature, Q (heat transfer to or from a system) = 0
Adiabatic expansion can occur in a wellinsulated system. Neglecting kinetic energy, electrical energy, etc, the drop in enthalpy of the system is effectively converted to work (dH = Q + W, where Q = 0). An adiabatic expanion is thus considered to be most expanion that can occur.
Adiabatic Efficiency is the ratio of the actual work output of the engine to the work output that would be achieved if the process between the inlet state and the exit state was isentropic (see below).
Isentropic (or Isoentropic) Process and Isentropic Efficiency
An isentropic process is an ideal or “perfect” process in which entropy remains constant. For a reversible isentropic process, there is no transfer of heat energy and therefore the process is also adiabatic.
If a process is both adiabatic and reversible, then it is considered to be isoentropic.
Isentropic expansion of steam is represented as a vertical line on a Mollier hs diagram. However expansion of steam is never perfect and some increase in entropy cannot be avoided. Real expansion is represented by a sloping (nonvertical) line, the angle of slope being indicative of the isentropic efficiency of the expansion.
In the case of a steam locomotive, the isentropic efficiency of the expansion of steam in the cylinder is found by dividing the specific work done in the cylinder by the isentropic heat drop between admission and exhaust. It therefore defines the efficiency of the engine unit (i.e. cylinders, valves etc) in terms of the amount of work it delivers from each stroke compared to the isentropic heat drop of the steam between admission and release.
Isentropic efficiency is therefore a measure of the efficiency of the engine unit. As with a locomotive’s thermal efficiency, its isentropioc efficiency varies with speed, cutoff, steamchest pressure etc.
An example appears in the 5AT FDCs, where lines 68 to 84 of FDC 1.3 are used to calculate the isentropic efficiency of the 5AT at maximum drawbar power.
An example of isentropic efficiency being applied in a calculation can be found here.
Isenthalpic (or Isoenthalpic) Process
An isenthalpic process is one that proceeds without any change in enthalpy (H) or specific enthalpy (h). There will usually be significant changes in pressure and temperature during the process.
In a steadystate, steadyflow process, significant changes in pressure and temperature can occur to a fluid. However the process will be isenthalpic if
 there is no transfer of heat to or from the surroundings (i.e. it is adiabatic),
 there is no work done on or by the surroundings, and
 there is no change in the kinetic energy of the fluid.^{ }
The throttling process is an example of an isenthalpic process – for instance the lifting of a safety valve on a steam boiler. The specific enthalpy of the steam inside the boiler is the same as the specific enthalpy of the steam as it escapes from the valve. Thus with a knowledge of the specific enthalpy of the steam and the pressure outside the pressure vessel, it is possible to determine the temperature and speed of the escaping fluid.
In an isenthalpic process: h_{1} = h_{2} and therefore dh = 0.
The above definition comes from Wikipedia and also World Lingo which offers an almost identical definition. However neither is entirely satisfactory since steam escaping through a safety valve will experience rapid cooling in its surroundings. The diagram below (from Chemical and Process Technology) gives a clearer picture even if the accompanying explanation is less so.
An interpretation of the accompanying explanation is offered as follows:
The flow through a pressure relief valve is extremely fast. Choked flow can occurs as far as position A inside the nozzle. The flow from the inlet to “A” will be a REVERSIBLE process and thus an ISENTROPIC process. Beyond “A” to the outlet of the valve, the steam expands (it may even undergo a change in state if it is liquid prior to “A” resulting in a transformation energy loss) followed by a rapid loss of speed and conversion of kinetic energy to mechanical energy in the form of noise. However the enthalpy remains constant (ISENTHALPIC). This process is IRREVERSIBLE (i.e. entropy increases).
Heat Capacity Ratio, Isentropic Expansion Factor, or Expansion Coefficient
[Ref Wikipedia] The Heat Capacity Ratio is sometimes
also known as the “‘isentropic’ expansion factor“. In the 5AT FDCs, the expansion factor is termed the “expansion coefficient” and denominated by the letter ‘n’. In other texts it may be denoted by γ, κ or the letter k.
The value of n is derived from the equation: n = C_{p} / C_{v} where, C is the heat capacity of a gas, suffix P and V refer to constant pressure and constant volume conditions respectively.
The heat capacity ratio (expansion coefficient) ‘n’ can be visualized from the following experiment:
A closed cylinder with a locked piston contains air. The pressure inside is equal to the outside air pressure. This cylinder is heated to a certain target temperature. Since the piston cannot move, the volume is constant, while temperature and pressure rise. When the target temperature is reached, the heating is stopped. The piston is now freed and moves outwards, expanding without exchange of heat (adiabatic expansion). Doing this work cools the air inside the cylinder to below the target temperature. To return to the target temperature (still with a free piston), the air must be heated. This extra heat amounts to about 40% more than the previous amount added.
In this example, the amount of heat added with a locked piston is proportional to C_{V}, whereas the total amount of heat added is proportional to C_{P}. Therefore, the heat capacity ratio in this example is 1.4.
From Steam Tables the following outputs can be found (based on a pressure of 20 bar)
Pressure = 20 bar  200^{o}C  300^{o}C  400^{o}C  450^{o}C  
Cv  kJ/(kg.^{o}C)  2.06657  1.696  1.66439  1.67895 
Cp  kJ/(kg.^{o}C)  2.98955  2.32035  2.2013  2.19664 
n  –  1.35  1.37  1.32  1.31 
The coefficient is usually given the value 1.3 in SGS locomotive performance calculations.
Reversible Process
A reversible process is a process that, after it has taken place, can be reversed and causes no change in either the system in which the process takes place, and causes no change in its surroundings. In thermodynamic terms, a process “taking place” would refer to its transition from its initial state to its final state. Or put another way, a reversible process changes the state of a system in such a way that the net change in the combined entropy of the system and its surroundings is zero.
Since all processes involve some increase in entropy (however small), the concept of reversibility is an ideal one that can never be achieved in practice. However the concept of reversible processes is a useful one. For instance, it is used to define the maximum possible efficiency (the isentropic efficiency) of a heat engine in that a reversible process is one where no heat is lost from the system as “waste”, and the machine is thus as efficient as it can possibly be (see Carnot cycle).
A reversible process is thus of necessity an isentropic process.
Specific Volume: Specific volume of a gas is the inverse of its density. It is therefore a measure of volume per unit of mass. In steam tables, its units are usually given as m^{3}/kg (or lb/ft^{3} in the old units).
Page Under Development
This page is still “under development”. Please contact the webmaster@advancedsteam.org if you would like to help by contributing text to this or any other page.
Thermodynamics Nomenclature:
T = temperature (^{o}K) 
V = volume of system (cubic metres) 
P or p = pressure at the boundary of the system and its environment, (in pascals) 
W = work done by or on a system (joules) 
Q = heat transfer in or out of a system (joules) 
q = specific heat transfer in or out of a system (joules per kg) 
U = internal energy of a system mainly contained in solid and liquid components (joules) 
u = specific internal energy of a system (joules per kg) 
H = enthalpy of a system (joules) 
h = specific enthalpy (joules per kg) 
S = entropy (joules per ^{o}K) 
s = specific entropy (joules per kg per ^{o}K) 
n = heat capacity ratio (= C_{v}/C_{p}) 
Thermodynamics Equations:
Thermodynamics equations can be difficult to understand. The following is a simpified summary where the term “system” can be equated to a steam locomotive’s cylinder:
The First Law of Thermodynamics (conservation of energy) can be expressed as “The increase in internal energy of a system = the heat supplied to the system minus the energy that flows out in the form of Work that the system performs on it environment” [ref Wikipedia]. In this case, the external “environment” is the locomotive’s piston, hence the definition can be formulated by the equation:
δU = Q – W_{piston} ………… (1)
which may also be written:
dU = dQ – dW_{piston} ………. (1a)
However, the work done on a system (locomotive cylinder) by changing its volume is dW = p.dV, hence:
dU = dQ – p.dV …………. (1b)
If the process (steam expansion) is assumed to be adiabatic – i.e. with no heat transfer in or out, then dQ = 0, whence
dU = – p.dV ………………. (1c)
However Enthalpy (see separate page) is defined as the sum of a system’s internal energy plus the product of its pressure and volume – i.e.
H = U + P.V ……………..(2)
from which a change in enthalpy can be defined (by differentiation) as
dH = dU + p.dV + V.dp ……………..(2a)
Thus by combining equations (1c) and (2a) we get (for adiabatic expansion): dH = p.dV + p.dV + V.dp, or
dH = V.dp ……………..(3)
Combining eqns (1c) and (3) gives:
dH/dU = – V.dP / P.dV …………….. (4)
The Second Law of Thermodynamics (heat always flows to regions of lower temperature) can be expressed as “a change in the entropy (S) of a system is the infinitesimal transfer of heat (Q) to a closed system driving a reversible process, divided by the equilibrium temperature (T) of the system” [ref Wikipedia]. This definition is formulated by the equation:
dS = δQ/T or dQ = T.dS ………………(5)
By combining Eqn (4) with (1b), a change in internal energy is given by:
dU = T.dS – p.dV ………………………….(6)
Ideal Gas Laws (from physics): The ideal gas law is defined by the equation:
pV^{n} = k …………………………. (7)
where n is the “heat capacity ratio“: n = C_{p} / C_{v} = – V.dp / p.dV [ref Wikipedia]
Thus from equation (4):
n = dH/dU
———— page in progress as at 10th Mar 2011 ———–
Enthalpy is a “term of convenience” that is useful in the interpretation and application of Thermodynamics. Enthalpy is basically a measure of energy, but its main function is in the calculation and measurement of “flow energy” or gaseous energy (e.g. steam energy).
[Note (borrowed from mathpages.com): Enthalpy is not a specific form of energy. It is just a defined variable that often simplifies calculations in the solution of practical thermodynamic problems.]
The term enthalpy is defined as the sum of a system’s internal energy plus the product of its pressure and volume, or
H = U + P x V

where H is the enthalpy of the gas (in Joules),

U is its internal energy (in Joules),

P is its pressure in Pascals, and

V is its volume in cubic metres.
Put another way, Enthalpy = Internal Energy + Flow Energy.
Commonly a “system” may be a combination of solids, liquids and gases, in which case most of its internal energy applies to the solid and/or liquid components while the PV term defines the energy of the gaseous component. This is confirmed by looking at steam tables which show that the enthalpy of water (in its liquid phase) is almost identical to its internal energy.
Specific enthalpy (usually denoted by the lowercase letter ‘h’) is the enthalpy per unit of mass, often measured in units of kJ/kg.
Measurement or calculation of a change in enthalpy is usually more meaningful than the value itself. For instance, the energy inputs required to raise and superheat steam can be estimated from the change in enthalpy between each step of the process. The first table assumes that cold feedwater is injected into the boiler before being heated; the second table assumes that the feedwater is preheated to 100^{o}C before entering the boiler (in both cases taken to be at a pressure of 20 bar = 2000 kPa):
Enthalpy Rise without Feedwater Heating 

1 Start with cold water  Enthalpy of water at 25^{o}C  105 kJ/kg  Increase 
2 Raise water pressure to 20 bar  Enthalpy of water at 20 bar and 25^{o}C  107 kJ/kg  2 kJ/kg 
3 Raise water temperature to 212^{o}C  Enthalpy of water at 20 bar and 212^{o}C  909 kJ/kg  802 kJ/kg 
4 Evaporate at same temp & pressure  Enthalpy of steam at 20 bar and 212^{o}C  2798 kJ/kg  1890 kJ/kg 
5 Superheat steam to 400^{o}C  Enthalpy of steam at 20 bar and 400^{o}C  3248 kJ/kg  450 kJ/kg 
6 Total of enthalpy rises from point 1 to point 5 = 3248 – 105 =  3143 kJ/kg 
Enthalpy Rise with Feedwater Heating 

1 Start with cold water  Enthalpy of water at 25^{o}C and 1 bar  105 kJ/kg  Increase 
2 Raise water pressure to 20 bar  Enthalpy of water at 25^{o}C and 20 bar  107 kJ/kg  2 kJ/kg 
3 Raise water temperature to 100^{o}C  Enthalpy of water at 20 bar and 100^{o}C  418 kJ/kg  311 kJ/kg 
4 Raise pressure to 20 bar  Enthalpy of water at 20 bar and 212^{o}C  909 kJ/kg  491 kJ/kg 
5 Evaporate at same temp & pressure  Enthalpy of steam at 20 bar and 212^{o}C  2798 kJ/kg  1890 kJ/kg 
6 Superheat steam to 400^{o}C  Enthalpy of steam at 20 bar and 400^{o}C  3248 kJ/kg  450 kJ/kg 
7 Total of enthalpy rise from point 1 to point 6 = 3248 – 105 =  3143 kJ/kg 
It can be seen from step 2 of this second table that preheating of the feedwater to 100^{o}C reduces the energy required from the firebox by around 10%, hence offering a potential nominal fuel saving of this amount. [Note: if preheat is obtained from exhaust steam, then the resulting loss of energy from the exhaust will result in a small loss in exhaust system performance and thus a slight increase in cylinder backpressure and loss of cylinder efficiency. This loss, however, is far outweighed by the overall efficiency gain from feedwater heating.]
It can also be seen from both tables that the additional energy required to superheat the steam is small compared to the amount required to boil the water. The advantage of using superheated steam is that all of the energy in the steam can be put to use in the cylinder provided the degree of superheat is high enough to prevent the occurence (momentary or otherwise) of condensation in the cylinder.
For further information on the subject of enthalpy, see:
Entropy
Draft Text Only – Readers’ suggestions and inputs are welcome
When a car or locomotive runs or brakes, we say that the heat escaping from it is “lost”. The Second Law of Thermodynamics explains this by saying that “energy will tend to dissipate from a hot or high energy body to a cold or low energy sink”. However since the First Law states that “energy cannot be created or destroyed”, then the “lost energy cannot simply disappear.
The concept of entropy was invented to account for this anomaly. This “waste heat” (or waste energy) dissipates into the atmosphere which acts as a heat sink, absorbing the energy without measurable increase in temperature. In so doing, the absorbed heat or energy effectively becomes “useless” (or wasted) to the extent that it cannot be put to further practical use.
Merriam Webster’s online dictionary offers three reasonably understandable definitions of entropy as follows:

a measure of the unavailable energy in a closed thermodynamic system that is also usually considered to be a measure of the system’s disorder, that is a property of the system’s state, and that varies directly with any reversible change in heat in the system and inversely with the temperature of the system; broadly : the degree of disorder or uncertainty in a system;

the degradation of the matter and energy in the universe to an ultimate state of inert uniformity;

a process of degradation or running down or a trend to disorder.
A better defintion comes from mathpages.com which offers the following: “The property that we call entropy is a measure of the uniformity of the distribution of energy”. It goes on to state that the entropy of a system equals the sum of the entropies of its component parts.
In simplistic terms, entropy can be thought of as a means of quantifying degraded or “useless” energy. The flame in a locomotive firebox contains a high concentration of energy (enthalpy) and a relatively low entropy, while the (lower temperature) steam in the boiler has a lower enthalpy and greater entropy. Exhaust steam leaving the chimney has an even lower enthalpy and higher entropy.
In fact, contrary to Merriam Webster’s definition, entropy is not a measure of energy. This may be deduced by the fact that by definition, the entropy of a closed system (e.g. the universe) gradually increases over time and is therefore not conserved. Entropy is in fact defined in units of energy per unit of temperature, as per the equation S_{1} – S_{2} = δQ/T where δQ represents a (small) amount of heat transfer to or from a body, T is its temperature and S_{1} and S_{2} its entropy before and after the energy transfer. The equation can be written dS = dQ/T giving entropy in units of energy per unit of temperature.
Specific entropy is the entropy of a system (usually a gas) divided by its mass, or in other words its entropy per kg in units of Joules per kg per ^{o}K.
In practical steam locomotive terms, values of specific entropy are usually found by looking them up in a set of steam tables or deriving them from a steam table program.
For further information see:
There are six interrelated properties that define the state of steam:
 Temperature
 Pressure
 Dryness Fraction (within the saturated zone)
 Enthalpy
 Internal energy
 Entropy
Fixing the value of any two properties defines the value of all the others. Thus fixing the values of Enthalpy and Entropy is sufficient to define Temperature, Pressure and Internal Energy of the steam.
The term “Mollier diagram” (named after Richard Mollier, 18631935) refers to any diagram that features Enthalpy on one of the coordinates. Commonly used Mollier diagrams are the enthalpyentropy (or hs) diagram (below) and the pressureenthalpy diagram illustrated at the bottom of this page.
The EnthapyEntropy or hs diagram:
The hs diagram is one in which Enthalpy values form the vertical axis and Entropy the horizontal axis. The values of the other related properties may be superimposed in the form of supplementary curves.
In the diagram below:
 green lines show steam temperature;
 blue lines give (absolute) steam pressure; and
 red lines give the dryness fraction (in the saturated zone).
Ideal (isentropic) expansion is represented on the Mollier diagram by a vertical line. Actual expansion of steam always involves some losses represented by an increase in entropy.
In the 5AT FDCs, Wardale gives examples of this, for instance in lines 68 to 84 of FDC 1.3 where he calculates the isentropic efficiency of the 5AT at maximum drawbar power. Here he assumes that the steam enters the cylinders at an absolute pressure of 21.39 bar and temperature of 450^{o}C shown as point A in the diagram below.
He then assumes that the steam will exhaust at an absolute pressure of 1.5 bar. Thus an isentropic expansion line AB can be drawn vertically at a constant entropy of 7.254 kJ/K kg, with point B being defined by pressure = 1.5 bar.
However in FDC 1.3 line 78 Wardale calculates that the actual exhaust steam enthalpy = 2,803 kJ/kg which allows point C to be located on the 1.5 bar pressure line. Thus the actual steam expansion is defined by the line AC.
The slope of the line AC is indicative of the isentropic efficiency of the expansion: the nearer the line is to vertical, the higher isentropic efficiency. Actual isentropic efficiency is determined by dividing the specific work done in the cylinder (FDC 1.3 line 82) by the isentropic heat drop between admission and exhaust (FDC 1.3 line 83).
Porta used similar lines in his Compounding paper but has simplified them by omitting
all the irrelevant lines from his diagrams.
The PressureEnthalpy diagram ilustrates the relationship between pressure and enthalpy with curves representing temperature (T), entropy (s), specific volume (v), and dryness fraction (x).
The domeshaped curve represents the limits of saturation – within the curve, steam is saturated.
Note: The same points A, B and C from the diagram above are illustrated in red. In this case the constant pressure line BC is straight and the constant entropy line AB is curved.
Anomalies in Thermodynamics
Draft Text Only – Readers’ suggestions and inputs are welcome
Thermodynamics is not an easy subject to master and there are many instances where it defies intuitive judgement. The following examples are offered by way of enlightenment and perhaps education:
 Steam viscosity: it is intuitive knowledge that liquids become less viscous as they get warmer. Oil becomes thinner as it is heated and syrup gets runnier. However the opposite is true for gases like steam. The higher the temperature of superheated steam, the higher its viscosity becomes. The following examples are taken from steam tables:
Steam Pressure bar (psi) 
Steam Temperature 
Steam Viscosity 
17.5 (250)  205 (saturated)  15.9 
17.5 (250)  300  20.1 
17.5 (250)  400  24.4 
The viscosity of gases increases as temperature increases and is approximately proportional to the square root of temperature. This is due to the increase in the frequency of intermolecular collisions at higher temperatures (reference: http://www.thenakedscientists.com/forum/index.php?topic=23433.0).
 Enthalpy reduces with increasing pressure: A more challenging and less easily explained anomaly is the fact that at constant temperature, the enthalpy and internal energy of steam falls with increase in pressure.
In his book “HushHush: The Story of the LNER 10000“, William Brown quotes from an unidenfied report written circa 1930 that seemed to question the wisdom of high pressure steam in such locomotives as Balwin’s high pressure compound No 60000 and Gresley’s high pressure compound No 10000 of 1929. The report stated that: “if the (steam) pressure is increased while the temperature remains constant, the superheat and heat content per pound of steam fall off as the pressure is increased. If steam of 200 psi and of 350 psi is expanded from the same temperature under such conditions that the exhaust steam escapes at the same pressure and temperature and with the same heat content in both cases, it follows that the heat taken from the steam in the cylinders and converted into mechanical work will be slightly less with the high than with the low pressure steam. That is, with the same heat content in the exhaust steam, the higher pressure will not give greater thermal efficiency.”
The following examples taken from steam tables illustrate the point:
Steam Temperaure deg C 
Steam Pressure 
Internal Energy 
Enthalpy kJ/kg 
300  13.8 (200)  2786  3041 
300  24.1 (350)  2764  3012 
Wardale‘s explanation of this phenomenon is instructive (quoted below):
The premise is that steam expanded from high pressure will produce less work than that expanded from a lower pressure and the same temperature if both exhaust at the same pressure and temperature. This would be true, but if comparing likeforlike, which is essential for the given conclusion to be made, the mistake arises because they will not exhaust at the same temperature, and the exhaust steam will therefore not have the same heat content in both cases. This is demonstrated in the following table, taking your own figures for temperature and pressure. (Note two things: steam properties are always given for absolute pressure, not gauge pressure, essentially because steam can (and should, wherever possible, i.e. power station turbines) be expanded to below atmospheric pressure (but don’t even think of it for a locomotive, please), and it is enthalpy we must use as the measure of steam thermal energy, not internal energy because as soon as a piston moves flow work is involved (consult a thermodynamics text book).)
From the enthalpy – entropy chart (Porta’s ‘Mollier diagram’) we have as follows:
Inlet Pressure, psi gauge 
200 deg C 
300 deg C 
Inlet pressure, bar abs.  14.81  25.15 
Inlet temperature, deg C  400  400 
Inlet steam enthalpy, kJ/kg  3256  3239 
Inlet steam entropy, kJ/kg.deg C  7.277  7.014 
Exhaust pressure, psi gauge  10  10 
Exhaust pressure, bar abs.  1.7  1.7 
For an isentropic = constant entropy or ideal expansion in both cases:  
Exhaust steam entropy,kJ/kg.deg C  7.277  7.014 
Exhaust steam enthalpy, kJ/kg  2731  2633 
Heat drop = work done, kJ/kg  (3256 – 2731) = 525  (3239 2633)= 606 
Increase in work done for expansion from higher pressure: 
((606 – 525) ÷ 525) x 100% = 15.4% 
If the same isentropic efficiency (say 85%) is applied to both cases to give a practical expansion allowing for incomplete expansion loss, etc., the relative values always stay the same.
This gives the correct situation and shows the book’s premise to be wrong because it is based on the false assumption of identical exhaust steam condition for both expansions. However the actual conditions are subject to various factors, making a generalized statement rather meaningless, amongst them:

 The above isentropic expansion from 200 psi gauge to 10 psi gauge gives an exhaust steam temperature of about 131°C, which is still superheated. However the corresponding expansion from 350 psi gauge gives saturated steam at 116 °C and dryness fraction 0.972, hence condensation losses at the end of this expansion would be high. But the entropy rise associated with imperfect expansion acts to increase the exhaust steam temperature for all expansions and hence reduce any tendency for condensation.
 It is unrealistic to have both expansions start from the same temperature. As the saturation temperature of steam at 350 psi gauge is about 30oC higher than that at 200 psi gauge, if both were superheated to the same temperature the degree of superheat (i.e. the amount by which the superheat temperature is in excess of the saturated temperature) would be that much lower for the former. Rather, as steam at higher pressure would enter the superheater at higher temperature, so the inlet steam temperature at the cylinders would be somewhat higher. This would decrease the tendency for condensation at the end of the expansion.
 Simply comparing ultra high with normal inlet pressure without specifying the nature of the engine (i.e. simple or compound) is misleading, compound being more suited to the former, simple to the latter. In the case of a compound, the superheat temperature must be set higher as the greater expansion possible in a compound (giving a higher isentropic efficiency, a fundamental reason for compounding) would otherwise result in saturated exhaust steam.
Steam Tables
Steam tables used to be (and may still be) published in the form of a small booklet listing values of steam temperature, pressure, specific volume (i.e. the volume of 1kg of steam at a given temperature and pressure), specific entropy, specific enthalpy against various input values, separate sets of tables being provided for saturated and superheated steam.
Nowadays it is easier to find values of enthaly and entropy (and many others) using a computer program, of which many are freely available for download from the internet, a good example being:

SteamTab™ Companion from http://www.chemicalogic.com/ producing outputs for both saturated and superheated steam in either metric and imperial units.

Calcsoft.zip from http://www.winsim.com/steam/steam.htmll, a DOS based program that works only in imperial units. However it offers additional functionality that allows entry of steam inlet and outlet conditions in order to calculate power output, isentropic efficiency and various other values.