Rated Tractive Force
EB11/R explains how to calculate the average tractive force (in pounds) exerted at the rail by the driving wheels of a two-cylinder steam locomotive engine: it is the product of the mean effective pressure (p.s.i.) of steam in the acting cylinder, the square of the piston diameter (inches), and the length (inches) of the piston stroke, divided by the diameter of the wheel. (See also Ludy, 131). The mean effective pressure at start-up is usually assumed to be 85% of the boiler pressure.
In this article, when I refer to "cylinder diameter," what I really mean is the size of the bore, which is, roughly, the piston diameter. Also, I may express the cylinder bore diameter and piston stroke length in shorthand form as, e.g., "16X24" (16 inch bore, 24 inch stroke).
The drawbar pull—which determines the load that the locomotive can haul—is its tractive force at the rail, less the resistance imparted by the locomotive itself.
The steam locomotive develops the rated tractive force made possible by its boiler and engine only if it can adhere to the track.
Maximum (Adhesion-Limited) Tractive Force
The effective tractive force applied to the wheel rims cannot exceed the "adhesion," which is the product of the weight which the locomotive places on its driving wheels, and the "coefficient of adhesion." This coefficient (Armstrong and others use 0.25; EB11/R, 0.2.) expresses how well the wheels resist sliding on the rails; higher is better. The engine can apply more force to the wheels, but they will just slip, not turn.
Consider the rail-riding pickup truck on the Grantville-Halle line. We are probably talking about a 200-300 horsepower engine, and a vehicle weight of around 5,000 pounds. At 10 mph, that engine could develop a pull of 9,000 pounds—if only the wheels didn't slip. But its maximum tractive force, thanks to the adhesion limit, is just 1,250 pounds. That means that it is way over-muscled, relative to its adhesion. Of course, "its muscles are designed for rubber tires on pavement, which have a much higher coefficient of friction." (Douglas Jones comment)
Designing for Adequate Traction
The desired tractive force can be calculated if we know how much tonnage the locomotive must move, over what grades and curves.
We then first ensure that the weight on the drivers is sufficient to generate adhesion at least equal to that desired tractive force. We distribute this weight across a sufficient number of axles so that the rail can handle the load.
Next, we must size the engine and boiler so that the rated tractive force is sufficient and sustainable. A logical starting point would be to use the design parameters of an old timeline (OTL) locomotive. EB11/R provides some useful comparative data for thirty-six different locomotives: wheel configuration, the position (inside or outside), diameter, and stroke length of the cylinders; the diameter of the driving wheels, the weights of the engine and its tender; the weight carried by the driving wheels; the grate area and total heating surface of the firebox. In fourteen cases, it also states the boiler pressure.
While EB11/R has something of a British bias, Alexander provides significant design parameters for over fifty American locos.
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It should be evident from the discussion of rated tractive force that this can be increased (up to the "adhesion limit") by
1) increasing the mean effective pressure (usually by increasing the boiler pressure),
2) increasing the cylinder diameter or the piston stroke length, or
3) decreasing the driving wheel diameter
Decreasing the drive wheel diameter is the only way of increasing long-term tractive effort which does not require that the boiler and firebox be enlarged to pay for it. However, it, too, has a price: reduced speed.
Large wheels are reserved for express passenger service, while small wheels are used on freight locomotives to maximize tractive effort. But the wheel diameter cannot be made too small, because it must remain larger than the piston stroke length.
For a freight locomotive, 42 inch driving wheels are typical. On a general purpose locomotive, a typical wheel diameter might be 54 inches. A little more speed, a little less tractive force. For a dedicated passenger locomotive, wheel diameter is likely to be in the 60-90 inch range, resulting in a more pronounced tradeoff of hauling ability for speed.
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If you want to increase the boiler pressure, you will have to evaporate water at a faster rate. This will require various firebox and firetube modifications. And to contain that pressure, you will have to use thicker boiler and firebox walls, which will make the locomotive heavier.
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If you increase the cylinder diameter or piston stroke length, you increase the engine's steam requirements. If the boiler is mismatched to the engine, then the boiler pressure will drop, the engine will gulp for steam, and the tractive force will decline. So changes in the engine ultimately affect boiler and firebox design.
Increasing the cylinder diameter or the boiler pressure also increases the force on the piston so the piston rod must be made larger and heavier to withstand the stresses imposed by piston motion. Which in turn affects the size of the main rod, the coupling rods, the axles, and even the frames.
Making the reciprocating parts (e.g., piston) more massive will increase shaking, which will mean more wear and tear on the engine, the running gear, the wheels and even the rail.
Increasing the stroke length necessarily increases the length of the piston rod, and hence its diameter must be increased so it doesn't buckle when compressed. The rest of the running gear then needs to be scaled up, too. With the same consequences as before.
Any mechanical engineer (there are at least ten in the Grid) will have studied, and will have textbooks describing, the basic mechanics of columns and beams, crank-and-rod mechanisms, etc. and hence will appreciate the mechanical limitations on piston rod length and cylinder diameter. It is no doubt because of the forces at work that cylinder diameters and piston strokes on locomotives rarely exceed 30 inches, even though a more massive design would increase tractive force.
Weight and Size
Because of the role of adhesion weight, lightening a locomotive is not necessarily advantageous. Indeed, in 1835 Baldwin built the first locomotive (The Black Hawk) in which the tender was integrated into the locomotive body, so that part of its weight would contribute to the traction. (Alexander, 50).
A locomotive is likely to be designed so that the weight it places on its drivers is at least four times the desired tractive force. Of course, its boilers and engines should then be sized accordingly.
However, the basic rolling resistance of a locomotive is still proportional to its total weight. There is substantial additional resistance, again proportional to total weight, when the locomotive steams up a slope, or accelerates.
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We also need to consider wheel weight. The greater the weight, the greater the wear on the rail, and the risk of rail failure.
The quality of the track is the principal limit on wheel load. EB11/R (847) says that a weight of 37,000 pounds "could be easily carried on one axle," and that implies 19,500 on each wheel. The heaviest rail mentioned by EB11/R is 100 pounds per yard, so a conservative rule of thumb would be to allow a wheel load of 195 pounds per pound of rail weight.
A contemporary Baldwin Locomotive Company catalogue states that if steel rails are properly supported by cross-ties, they can support a maximum wheel weight of 225 to 300 pounds for each pound per yard of rail. Thus, if a rail is dimensioned so that its weight is forty pounds per yard, no more than 12,000 pounds weight should be placed on a single wheel.
For a given weight on the drivers, wheel load can be reduced by increasing the number of driving wheels. In OTL, there was an increase in the number of coupled driving wheels.
One 1893 locomotive (Alexander PL87) had 84,000 pounds on four driver wheels, and thus the individual wheel load was 21,000 pounds (suitable for 70 pound or heavier rail). In contrast, another (plate 90) had 172,000 pounds on the drivers, but it w
as spread over ten wheels, and thus it could actually run on lighter track.
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The wheelbase is the distance from the first driven axle to the last one. If the normal axle spacing and wheel diameter are maintained, increasing the number of driving axles lengthens the wheelbase, which makes it more difficult for the locomotive to handle a curve. (Clarke, 122). Or turn around in a turntable or wye.
If the wheelbase is made too short, the locomotive becomes unsteady at high speeds. This was a problem with four-wheeled locomotives. (Clarke, 112-3).
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There are constraints on height and width, too. The so-called "loading gauge" (the clearances provided by bridges, tunnels, road cuts, stations and neighboring track) comes into play here. In America, the rolling stock can be as wide as 10'10" and as high as 16'2." (NOCK/RE, 208-9).
The width is constrained, not only by the loading gauge, but also by the track gauge (the distance between the inside edges of the rails), as a large vehicle on a narrow gauge track may tip over when running a curve. The standard American track gauge is 4'8.5."
Likewise, the height not only cannot be so great as to be "clipped" by the roof of a tunnel, it cannot be disproportionate to the width, or the locomotive will topple over.
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Increases in the dimensions of the locomotive will ordinarily mandate an increase in weight, too, unless a new, lighter structural material is employed. The materials presently available to the USE are wood, cast iron, wrought iron, steel, and a few other metals such as copper.
In nineteenth-century America, wood was used mostly in the cab and the tender frame, and as insulation. Copper was sometimes used for the heat exchange elements, because it conducts heat well, but it is structurally weak and thus copper tubing is thicker than the steel equivalent. Cast iron was used in cylinders, journal boxes, and valve boxes. For all other major components, the initial preference was for wrought iron, but this changed once the Bessemer process (1856) made steel affordable. By 1900, virtually the whole locomotive was made of steel. (White, 29-31).
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We cannot put into a locomotive the most powerful boiler and the most powerful engine available, only those whose power is greatest within weight and size constraints. And the engine and boiler compete for the mass and volume allotted.
Making Steam: Locomotive Boiler Design
The boiler is the stomach of the locomotive. It consumes fuel, air and water, and belches steam. The fuel is burnt to change chemical energy into heat energy; the air is necessary for combustion to occur, and the water is what is heated to generate steam. It is the expansion of steam which moves the pistons, and ultimately makes the wheels go round.
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Coal is shoveled onto a horizontal grate in the firebox, which receives air from the "ashpan" below, as well as, intermittently, through the firebox door.
The first fireboxes were mounted "inside" the wheel lines, and were long and narrow (grate area 17-18 square feet). Later, they were placed on top of the frame, and were wide but short (30 square feet). Long, wide fireboxes (up to 90 square feet) were made possible by relocating them behind the driving wheels. (Forney; Bruce, 36-43)
The smoke puffing from the steam locomotive is photogenic, but it is also evidence that fuel is being wasted. In 1859, engineers solved this problem with two new elements, a brick arch and a deflector plate. Together, they controlled the airflow so as to improve combustion.
"Monty" should be familiar with these two firebox features.
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There are two basic methods of using the released heat energy. Most railroad boilers were of the "fire tube" type, which means that the hot air rises from the coals and enters a multitude of pipes. These travel through the main section of the boiler, which holds the water. The heat brings the water to a boil, and the steam rises from the top of the water surface, ultimately collecting in the "steam dome." The fire tubes empty into the smoke box, and the smoke ultimately escapes through the smokestack. This creates a partial vacuum in the smoke box, which helps to draw in the air. EB11 "Boilers" shows two views of an express locomotive boiler (Fig. 10).
A few OTL locomotives were equipped with water tube boilers. Water is circulated in tubes through the firebox, rather than hot air through the water reservoir. Water tube boilers were much safer to operate, and potentially more economical, "but it was impossible to build efficient boilers of this type within the clearance limitations of the railway engine" (Sinclair, 691).
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The most efficient boiler operation is at a relatively low rate of combustion, e.g., 30-60 pounds of coal per square foot of grate per hour, resulting in evaporation of 11-13 pounds of water per pound of coal, and a boiler efficiency of about 80%. Burning 100-180 pounds per square foot of grate per hour, we obtain only about 6-8 pounds of water per pound of coal, and the boiler efficiency is about 40-50%. (EB11/R) Forney says that the most coal which can be burnt is about two hundred pounds per square foot of grate per hour, and then only at most six pounds of water would be evaporated by each pound of coal fired.
The size of the grate determines how much coal can be burning at one time. So a big grate seems like a good thing. However, there are problems of increasing its size. First of all, it means increasing the overall size of the locomotive. Secondly, once the grate exceeds a certain size, it becomes too difficult for a single "fireman" to keep it "fired" properly. (This was a problem with hand-fired "Pacific" locomotives, NOCK/RE 175.) You either need to provide two fire doors, for two firemen, or engineer a "mechanical stoker."
The firebox is positioned within the boiler so that there are water spaces to the sides and in back of the firebox, to maximize the direct firebox-to-boiler surface area (Alexander PL79). There is also water above the top of the firebox, the "crown sheet," and indeed the most common cause of a boiler explosion is that the crown sheet loses this protective blanket, and melts.
Heat transfer takes place not only at those walls of the firebox which are in contact with the water reservoir, but also at the walls of the tubes. So having lots of small diameter tubes is good—unless you are the fellow who has to make sure that those tubes are tight.
The longer the tubes, the greater the heat transfer area, but the weaker the combustion-promoting draft in the firebox. Having lots of tubes increases the heating area, but weakens the tube plate of the firebox.
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EB11/R discloses both the grate area and the total heating surface for 36 locomotive designs. Disregarding the Stephenson Rocket, the total heating surface ranged from ~1,400 to ~6,100 square feet, and the grate areas from 20 to 100 square feet. The average ratio was 71:1.
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The steam passes up into the steam dome, from which it is released to the cylinders by the throttle valve. Some locos had two steam domes, or other provisions for storing more steam.
The boiler pressure is a function of the rate at which steam is produced (evaporation rate), the rate at which steam is used, and the size of the steam reserve. Taking advantage of a large steam reserve to briefly make faster-than-normal speed or pull an extra-heavy load is called "mortgaging the boiler."
If you are producing a lot of steam quickly, the boiler pressure will increase. The pressure which the boiler can tolerate is dependent on the thickness of the walls, as well as the nature of its construction. Thicker walls can hold higher pressure steam, but the boiler will weigh more.
Alexander provides only limited boiler pressure data. An 1860 engine had 130 p.s.i. (PL47); locomotives built as late as 1882 had 125 p.s.i. pressure (PL76-7); three later locos were 180-190 (PL80, 85, 96). The highest pressure in the EB11/R table was 235.
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Bear in mind that since the cab is behind the boiler, a large boiler limits the crew's view of what is in front of them.
Putting Steam To Work: Locomotive Engine Design
Usually, the locomotive will have a pair of pistons, which operate one-quarter of a cycle out of ph
ase, so when one is in "neutral" the other is ready to receive steam.
The cylinders can be mounted outside or inside the main frame. In general, during the nineteenth century, the British preferred to use inside cylinders, and the Americans, outside ones (Nock/RE 164).
There are some locomotives which have a second pair of cylinders, in which case it is very common to have one pair on the inside and the other on the outside. However, both pairs can be on the outside. (EB11/R).
There was experimentation with other positions in the early days, but the cylinders of late nineteenth-century locomotives were mounted horizontally, and at axle level.
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Looking at the locomotive data in EB11/R, and ignoring both the primitive "Rocket," and engines with more than one pair of cylinders, we can see that the cylinders are 18 3/8-23 inches wide, and the piston stroke is 26-30 inches long. For the American locomotives in the Alexander book, if we ignore the pre-1840 models, cylinder diameter is 12-22 inches, and piston stroke 15-30 inches (save for one "13/54" locomotive).
For both the British and American locomotives, the stroke length was, on average, 50% greater than the cylinder diameter.
Cranking the Wheels: Locomotive Transmission Design
In the standard "rod" locomotive, the pistons are connected to cranks on the driving wheels, so two power strokes by the piston, make one turn of the crank, resulting in one revolution of the wheel.
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The driving wheels of a high speed locomotive may turn at a rate of more than five revolutions per second. During each half-revolution, each piston accelerates to full speed (say, 35 feet per second) and then decelerates to a full stop. The necessary force on it is the mass times the acceleration. The piston weighs, say, 500 pounds (Forney), and the maximum acceleration is proportional to the stroke length and the square of the wheel speed (EB11/SE 837). The piston transmits that force to the piston rods, cranks, and other elements. They all must be able to withstand the resulting stresses, and, unless they are balanced, they cause unpleasant, perhaps dangerous, vibrations in the locomotive structure.