### Introduction

There are a lot of myths and legends about locomotive power and the comparisons between steam and diesel locomotives, which have led to a lot of misconceptions and arguments. Many of these arguments are based on romantic ideas of the beauty of steam and the perceived characterlessness of diesel locomotives. However, in the end, railroads only want to purchase power which will haul the loads in the most efficient way. Diesels replaced steam locomotives because that's what they did - they are more efficient because they cost less money to run. This article, written by US locomotive engineer Al Krug in a series of newsgroup posts, tries to explain the power questions that show how diesels are more efficient than steam locomotives. Oh yes, sorry, the figures are in imperial measurements. The US does not use metric measurements yet.

### Steam Loco Physics

In any debate about the differences between diesel and steam locomotives the question of power comes into play. Whenever you are talking about the power of steam locos you must specify the speed, otherwise the statement is meaningless. Diesel Electric locomotives (DEs) develop their rated HP at any speed whereas steam develops it at only one speed. Why? Let us look at the physics.

We will assume a hypothetical 4-8-4 steam locomotive that weighs about 400,000 lbs without the tender and has 250,000 lbs of its weight actually carried by the driving wheels. The cylinders will be 28" in diameter (bore) and have a stroke of 30". The drivers are 60" in diameter and the boiler pressure will be 250 psi. All of these are "in the ball park" figures for a 4-8-4. We open the throttle and allow the 250 psi steam from the boiler to enter the cylinder. This steam pushes against the face of the piston. The force generated is the steam pressure times the surface area of the piston. In this case the area of the piston 14" squared times the constant "pi" or 14 x 14 x 3.14 which = 615 square inches. Each sq. inch of this is pushed on by 250 lbs (the steam pressure), which results in a total piston force of about 153,000 lbs. This force is connected to the driving wheel via the piston rod, main rod and driver crank pin.

The stroke of our engine is 30". This means that the crank pin must move 30" from front to back as it rotates about the driving axle as the driving wheel turns. That means the crank pin must be 15" away from the centre of the axle. The crank pin and driving wheel transmit the piston force to the rail. When the crank pin is at the point in its rotation directly below the driver axle, you can think of the section of the driver between the axle and the rail as a simple lever. The fixed end of the lever is at the axle and the free end is at the rail. The force is applied to this lever at the crank pin. The crank pin is 15" from the fixed end of the lever and the rail is 30" from the fixed end. This gives a 2 to 1 REDUCTION in the force applied at the crank. In other words the rail feels only 1/2 of the force applied to the piston. The piston produces 153,000 lbs of force and 76,000 lbs is felt at the rail.

### Limitations

There are several problems with assuming the full power of the loco can be applied to the rail, however. First of all we only have 250,000 lbs on the drivers. At 20% adhesion (probably high for a steam loco) this means we can only apply 250,000 x 0.20 = 50,000 lbs to the drivers before they will slip. So we can't use all that 76,000 lbs of thrust. Only 50,000 lbs of it.

At first glance this looks like we should not even bother trying to develop that amount of thrust to begin with. Perhaps we should lower the boiler pressure to 164 psi so we only develop e 50,000 lbs? Or maybe we should reduce the bore of the cylinder and thus the size of the piston to 22 inches so we only generate 50,000 lbs of thrust? Or we can increase the size of the drivers to 92"! That would limit the thrust reaching the rail to 50,000 lbs.

To use all the power available, we could increase the weight on drivers but not by much. We already have 62,500 lbs on each driving axle and 70,000 lbs is the maximum axle load limit i.e. the maximum the rail can hold. To use the full 76,000 lbs of thrust at 20% adhesion we would need 380,000 lbs on the drivers or 95,000 lbs per axle! Bend those rails! So, our only option to get increased weight on driving axles would be to add another driving axle and boost them all up to the 70,000 pound limit. This would give us 350,000 pounds on drivers, close to the needed 380,000 lbs. However we would still need to use one of the other methods mentioned above to reduce the remaining excess force. And, of course, we now have a 4-10-4 or a 2-10-2 not a 4-8-4.

In fact we do not have to do any of the things we have discussed to get the loco to work. What we have looked at so far has been some of the limitations you must consider and various ways of compensating for them. The real fact is that our original 4-8-4 loco will not deliver 76,000 lbs to the rail, actually only about 50,000 lbs. This is because we cannot apply full boiler pressure steam to the face of the piston throughout it entire power stroke. The AVERAGE cylinder pressure over the length of a stroke will actually be somewhat below the maximum boiler pressure.

There are several reasons for this. If we allow boiler pressure steam into the cylinder all the way to the end of this piston's stroke we are wasting steam. Think about it At the end of the stroke the crank pin is directly behind the driver axle. Pushing on the crank pin in this position does nothing to rotate the drive wheel. All the force does is try to bend the rod and shear the crank pin. So there is no sense at all in using steam to keep the cylinder pressure up near boiler pressure as the piston nears the end of its stroke. We may as well cut off the intake valve early and save this steam.

This means that the piston now no longer delivers its full force capability over its entire stroke, so our average thrust is considerably lower than the piston's maximum capability. The piston only applies its force for about an eighth of the driver's rotation, that portion when its crank pin is near the bottom of its rotational travel. This corresponds to the midstroke of the piston. The loco has four pistons however, two double-acting cylinders. The crank pin on the other side of the driver is 90 degrees off from this side so the power stroke of each piston supplies its thrust in a different quarter of the driver's rotation.

### Constant Force

At this point we have determined that our loco develops about 50,000 lbs of continuous pull. Note that this pull is constant regardless of speed. As long as we keep the same steam pressure against the faces of the pistons, the same force is produced regardless of the loco speed. A STEAM LOCOMOTIVE IS A CONSTANT FORCE MACHINE! That is a very significant fact. It is fundamentally different from the Diesel Electric locomotive, which is a constant HP machine. On the surface it would seem that a constant force machine would be great for a locomotive and indeed in some respects it is. Since the force or pull of a steam loco is constant regardless of speed what does that say about HP? Horsepower is the product of pull times speed (divided by a constant). If the pull is constant and the speed rises then the HP also increases. So you see, my original question asking "at what speed" is the HP developed was a fair question after all. If you quote steam loco HP you must also state the speed.

### Pulling Power

Lets couple our steam loco to a train. For our purpose the weight of the train does not matter as we are merely going to accelerate it on level track. A heavier train (more mass) will accelerate more slowly than a lighter train but the results will be the same. We open the throttle wide and instantly the steam loco develops its 50,000 lbs of pull. As the loco accelerates away from 0 mph its develops more and more HP. Here is a table of the theoretical HP of our constant 50,000 lb pull loco,

10 mph = 1333 HP

15 mph = 1999 HP

20 mph = 2666 HP

25 mph = 3333 HP

30 mph = 3999 HP

35 mph = 4666 HP

40 mph = 5333 HP

45 mph = 5999 HP

50 mph = 6666 HP

and so on. Man what a machine!

### Train Resistance

But in reality it does not work out that way. A train does not roll freely. At some speed the rolling resistance of our train will equal the 50,000 lbs of drawbar pull available and the acceleration will stop. According to the Davis formula for train resistance, a 50 car 2500 tons train will have 50,000 lbs of drag at 117 mph. At this speed the loco would be generating 15,600 HP. Obviously this does not happen, why not?

### Boiler Capacity

As loco speed increases so does the piston speed. The pistons make more and more power strokes in less and less time. Each power stroke requires steam. At some speed the appetite of the cylinders for steam is going to outpace the boilers capacity to supply it. The steam pressure is going to drop. Lower steam pressure means lower cylinder pressure which means lower piston force and thus lower loco pull. Since the HP of the steam loco is its pull times its speed the HP also levels off. The HP is limited by the capacity of the boiler to produce steam. We can remedy this by making the firebox bigger to burn more fuel and making the boiler bigger to hold more fire tubes to increase the heat exchange area. But now we have a bigger, different, loco. One that may be too heavy for 4 drive axles to support.

You can see from the above chart that if our loco is equipped with a 3,000 HP boiler it will begin to run out of steam at about 25 mph. That is as fast as the loco can go and still deliver its 50,000 lb drawbar pull. But it can go faster if we don't demand 50,000 lbs of pull from it. If steam pressure drops by half from 250 psi to 125 psi then the cylinder force is half and the loco pull is half. Now the loco has only 25,000 lbs of pull instead of 50,000. Allowing the boiler pressure to drop like this is bad practice. It is inefficient. It also affects the auxiliary devices such as electric light generator, air compressor and stoker engine. It would be much better if we kept the boiler pressure at 250 psi and limited the amount of steam we put into the cylinders. Instead of using the original amount of steam and causing boiler pressure to fall to half its original pressure, we can use half the amount of steam at full boiler pressure. The resulting force is the same.

### Cut Off

We can limit the amount of steam going to the cylinders by leaving the intake valves open for less of the stroke. We close the intake valves earlier in the stroke. We "cut off" the flow of steam to cylinders early. This is exactly what the cut off lever (sometimes called a Johnson Bar in the US and a Reversing Lever in the UK) does. When an engineer adjusts the cut off lever, "notches up the cut off", "hooks up the Johnson bar", he is limiting the amount of steam going to the cylinders. If we put half the amount of steam into the cylinders as before, then the average force on the pistons is going to be half of what it was before. This translates directly into half the pull as before. Now our loco is only delivering 25,000 lbs of pull at any given speed instead of 50,000. What this means is that our loco will continue to accelerate our train, albeit more slowly, until the product of the new lower drawbar pull times the new higher speed equals the 3000 HP boiler capacity once again. At this new, higher speed, the cylinders will be moving twice as fast as before but each stroke is only consuming half as much steam. The total steam usage is the same as before and our 3000 hp boiler can keep up. In the first scenario the loco began running out of steam at 25 mph. Now it does not begin to run out of speed until 50 mph.

Further limiting the steam flow to the cylinders means the loco will move even faster before the piston speed causes this reduced flow of steam to equal the boiler capacity. Of course the resulting pull will be even lower also. Once again, the product of this even lower pull and higher speed will equal 3000 HP, the capacity of our boiler to produce steam.

So this steam loco is not a pure CONSTANT FORCE machine. It is only a constant force machine from 0 to 25 mph. Above that speed we have to limit the force by use of the cut-off so we don't exceed the boiler HP capacity. Above 25 mph the loco is a CONSTANT HP machine just like the Diesel Electric. I said earlier, we could increase the size of the boiler and the loco would continue to develop more HP as it accelerates. This has limits other than the size and weight problem described earlier. Even if we have a boiler designed for tremendous HP there is the problem of getting all that steam into and out of the cylinders in an ever shrinking period of time.

### Back Pressure

The faster the loco goes the faster the pistons go back and forth in the cylinders and the less time we have during each stroke to move steam. At some speed the restrictions in the throttle valve, steam pipes, and valve passages are going to cause significant pressure drops. Pressure drop means less force and less HP developed. On the exhaust stroke we have to get all that steam out of the cylinder, through the exhaust valves, exhaust piping, and the exhaust nozzle. If we don't succeed in this, then some steam pressure remains in the cylinder, pushing against the wrong face of the piston and nullifying part of the steam pressure on the power side of this piston. This pressure is called back pressure. For these reasons we have to limit the amount of steam we try to put through the cylinders at high speed. Again the cut off lever comes in handy for this. So as speed increases we have to "notch her up", cut off the steam earlier in each stroke and HP levels off. There is no sense in putting a bigger boiler on this loco because we can't force any more steam through the cylinders anyway.

### An Example

Now that we have the fundamentals let's look at a real life loco, the Timken roller bearing 4-8-4 #1111. In tests on the NP it gave the following results (rounded off):

10 mph = 1400 hp

15 mph = 2100 hp

20 mph = 2600 hp

25 mph = 3000 hp

30 mph = 3000 hp

40 mph = 3000 hp

50 mph = 3000 hp

In its constant pull phase from 0 to 25 mph it produced 50,000 lbs of pull +/- 2000 lbs. At 50 mph it only produced 23,000 lbs. This real life loco follows very closely what I predicted.

*Figure
1: Northern Pacific 4-8-4 Class A-1 locomotive fitted with
demonstration Timken roller bearings as delivered in 1930. The number
1111 was temporary and led to the locomotive being nicknamed “The Four
Aces”.*

### Comparing Performance of Diesel Electric to Steam.

This Timken 4-8-4 loco #1111 is essentially a 3000 hp single unit locomotive. Let us compare its performance with that of an SD40-2. (or a two unit 3000 hp F7, it makes no difference as long as the total weight is the same). Our 4-8-4 weighs about 400,000 lbs total engine, not counting the tender. It has about 250,000 lbs of its weight on its drivers. Our SD40 also weighs 400,000 lbs. All of its weight is on its drivers. The steamer is a constant pull machine below 25 mph and a constant HP machine above that speed. The SD is a constant HP machine throughout its range of about 8 mph to max speed. What consequences does this have on train performance?

Assume we hook each loco to an identical train. This train has 50 cars and weighs 2500 tons and we are rolling on level track. Since both locos are essentially 3000 HP units they will perform identically at speeds above 25 mph. Using the Davis formula for train rolling resistance we find that this train will develop 21,000 pounds of rolling resistance at 53 mph. Multiplying these figures and dividing by the constant 550 ft-lb per HP gives us 3000 HP. This train will need 3000 HP of pull at 53 mph. So both locos will do this. Their performance is identical.

Next, we roll the trains onto a 0.75% grade. Now in addition to the rolling resistance of the train we also have grade resistance. The grade resistance of this train will be 37,000 lbs. Since our locos can only pull 21,000 lbs at 53 mph, our speed will drop. HP is speed times pull. Since both of our locos are constant HP machines in this speed range this means their pull will go up as they slow down. As the speed drops the rolling resistance also drops. At some speed, the sum of the lowering rolling resistance and the grade drag will equal the increasing pull of the locos. This occurs at approximately 23 mph. The grade drag will be 37,000 lbs and the rolling resistance will be 13,000 lbs for a total of 50,000 lbs. T he product of 50,000 x 23 mph = 3000 hp. Both locos will still perform identically.

Now let us roll onto a steeper grade, say 0.8% grade. On this grade the train has 40,000 lbs of grade drag. Its rolling resistance depends on its speed as usual. But at 25 mph it has 13,000 lbs of rolling resistance and 13,000 + 40,000 = 53,000 lbs. Our steam engine only has 50,000 lbs of pull. The train is going to slow down until the rolling resistance equals 10,000 lbs. (10000+40000=50000). This will occur at 5 mph. Our steam loco will chug over this hill with this train at a mere five mph. If you multiply the pull times the speed you find that the steam engine is only developing 666 hp! The DE on the other hand continues to develop its full 3000 hp at any speed. This means that as the DEs speed drops its pull continues to go up. At 21 mph the rolling resistance is about 12,000 lbs. The grade resistance is still 40,000 lbs. The total is 52,000 lbs and the DE can deliver this pull at that speed. 52,000 lbs times the speed is 3000 hp. (If you got lost in the math remember I am simplifying this and rounding off. The actual math is 52,000 lbs x 1.4666 ft per sec per mph x 21 mph / 550 ft-lb per sec per hp and that yields 2912 HP needed).

The SD will march this train over this grade at a whopping 21 mph! The steamer barely did it at 5 mph! What's more, if we roll onto a grade of 1.0% the grade resistance of our train is 50,000 lbs. This is all the steamer can muster, there is none left over for rolling resistance. The steamer is stalled. On a 1% grade our SD40 is still tromping along at 18 mph. In fact it will continue to pull this train on a grade as steep as 1.3% at a speed of 14 mph. At this speed the pull is 80,000 lbs. If the DE has a 20% factor of adhesion, the same used for the steamer, then it will slip if we try to pull more than 80,000 lbs (20% of 400,000 lbs = 80,000). So 14 mph is the slowest we can go at full throttle with the SD40. But the DE delivers its tractive effort in a much smoother continuous manner than the pulsing pull of the steam engine. Because of this it may achieve adhesion factors of around 25%. Using this figure we can load the unit down until it is pulling 100,000 lbs. For our fictional train above this works out to a grade of 1.8% and 11 mph.

From this discussion you can see that the DE beats the steam loco hands down. It is no contest at low speeds. The steamer is limited by its fixed drawbar pull. The drawbar pull is fixed because it is a function of boiler pressure, piston diameter and ratio of stroke to driver diameter. Increasing any of those values will allow our steam engine to generate more thrust. But we can not use it. The steamer is limited to 50,000 lbs pull by its 20% adhesion factor and its 250,000 lbs on drivers. Changing the weight on drivers creates an entirely different steam engine.

*Figure
2: Canadian Pacific Railway General Motors EMD SD40-2 diesel electric
locomotive. This was one of the most successful Americal diesel
locomotive designs ever built. It was in production from 1972 to 1989
and many still remain in service. *

### Boosters

One way we can get increased pull with out changing the wheel arrangement is to power the unpowered axles. We can add small steam cylinders to the trailing truck. Now at low speeds where there is excess boiler capacity we can use that extra steam in the trailing truck booster engine. This will increase the low speed pull because we have increased the weight on drivers by making non-driving wheels drivers. Booster engines on the trailing truck were maintenance headaches as I understand it. While the booster helps it still does not put all of the weight on the drivers. The leading truck is still unpowered so the steamer still cannot match the DE of same total weight.

### Increasing Weight

Increasing total weight is inefficient. The weight of the loco does not go along the track for free. It takes HP to move it just like the train. The weight of a loco is a compromise between the maximum pull you want and the efficiency. In our demonstration our SD40 was 7.8% of the total weight of the train. It will therefore absorb 7.8% of its own HP just moving itself. The steam engine will do the same thing. But steam engines are usually rated at DRAWBAR HP not cylinder HP. This means that the power to move the loco itself has already been subtracted when quoting drawbar pull. You still have to pay for the fuel to move the loco but its pull is not derated from this loss.

The DE on the other hand is rated at HP for traction. Therefore you must subtract the part needed to move the loco to get the HP available for the train. What this means is that the performance figures for the DE above will be roughly 8% less than stated. In spite of this, the DE still shines in comparison to the steam loco.

When it comes to HP efficiency to move the locos the DE again shines. The steam loco has a tender that when loaded can weigh as much as the steam engine itself. This means it takes twice the HP and fuel to self propel the steam loco & tender as for the DE. It is possible that the steam loco being an external combustion engine as opposed to the DE being an internal combustion engine uses its fuel more efficiently. That is it may extract more of the heat value of the fuel to do useful work. This would offset some of the efficiency loss of dragging around a dead weight tender. I do not know the relative combustion efficiencies of steam loco vs. DE. If anyone has data to show that I am loony and my physics is all wrong then by all means do correct me. I purposely rounded off and simplified many things to keep this from being any longer or more obtuse reading than it is.

From the discussion I have presented, I can draw only one conclusion.... The railroads should convert to diesel electric locomotives as soon as possible and the steam engines should all be cut up for scrap. Oh......wait.......they did that. (Yes I enjoy watching them, listening to them, and smelling them. But I sure don't want 'em on the point of my freights).

### Proof?

One may legitimately ask, why did I choose to compare the 4-8-4 to an SD40? The short answer is because it proves my point. Always beware of the writer's motives. The long answer is because the 4-8-4 (sans tender) and SD40 both weigh the same and both are 3000 HP. Our SD40-2s range from 383,000 lbs to 424,000 lbs. I have ran many many SD40s in my time so I am familiar with what they will or will not do. If I made a mistake in my physics logic or the math the results would have stood out. I would recognise that the figure was way out of line according to my knowledge of the SD40. If I had used the 2 unit F7 it would also be 3000 hp but it would weigh 500,000 lbs or more. All of that weight would be on drivers and it would make the steamer look even worse than the SD40 did. One may ask why not use a GP40 for the comparison. A GP40 is also 3000 hp like the 4-8-4 and like the 4-8-4 it also has about 250,000 lbs on drivers. (Actually GP40s I have been on are more like 270,000 lbs). A 250,000 lb GP40 would be in a dead heat with the 4-8-4 as far as train performance is concerned. The GP40 will slow down on the hill to the point where the drawbar pull has risen to 50,000 lbs just like the SD40 and the 4-8-4. As it continues to slow it will deliver more than 50,000 lbs of force to the wheels and it will slip. It would have to be throttled down to maintain the force at 50,000 lbs and thus becomes a constant pull machine just like the 4-8-4. The two locos are virtually identical in this respect.. The smoother force of the GP40 may enable it to get more than 20% adhesion, perhaps 25%. This would of course improve its performance somewhat over the steamer.

Another difference comes when you look at self-propulsion efficiency. The GP40 weighs 250,000 lbs verses 800,000 lbs for the steam loco with tender. The steam loco is going to take over 3 times the HP to self propel. The GP40 is about 60 feet long and the steamer is 110 feet. The problem with steam locos as far as physics is concerned is that old bugaboo, weight on drivers vs. total weight. I can't see them competing with DEs until you can build an 0-12-0 tank engine that will run at high speed or low all day & night without refuelling. All this and we still haven't addressed all those common complaints about steam vs. diesel electric.

### Other Points

You know, the fact that steam requires a fireman, is not MU-able thus requires a crew for each loco, needs frequent water stops, needs frequent maintenance, has imbalances which pound the track, must be turned, can't be left at outlying points, etc., etc. must all be taken into account. After all, it's value for money which counts.

Why compare an SD40 to a 4-8-4 steam loco. Because they were both the same HP and the same weight. Although to be really equal in the weight category I should have included the tender weight. Had I done that the steam loco would not have even gotten out of the starting gate in this comparison.

It is said that a DE's HP is engine HP not drawbar HP so a 3000 HP DE is not really the same as a 3000 HP steam loco. OK granted. So if I'm going to compare equal drawbar HP locos then I get to use a 3600 HP DE verses a 3000 HP steam loco. Well, this makes the DE even more favourable. However, it is my actual experience based on train tonnage, grade, and speed that DEs are putting out 88% of their rated hp just for the grade. If you apply the Davis formula to see what the train rolling resistance is at that speed and add that HP to the grade HP you find that the DE is putting out 95% of its rated HP to the drawbar. Granted this is at low speeds and will be somewhat lower at high speeds.

Some say I should have used an FT or F3 contemporary DE. Why? Steam had 100 years of evolution by then and the DE was brand new. I say at the least we should compare them after similar evolutionary periods. Some say I should have used an articulated loco like the Allegheny or Big Boy. In my humble opinion, an articulated loco is actually two steam engines permanently MU'd together. Some even had separate boiler sections and some were even connected by an accordion-like bellows. So, if I'd used an articulated loco then I should get to use at least a two unit DE of equal HP. "Oh no", they cry, "We are comparing only single unit locos." See my opinion above about articulateds. In any case then, since the articulated has two engines I shall use two engines. If I mount two GP40 engines, or two Dash9-44CW engines, or two SD80 engines on a single frame it will still beat out the articulated.

But why apply the artificial restriction of a single frame. If there was an advantage to doing that for a DE then it would be done. The DD35s and 40s were not repeated. If I have two engines/generators why not mount them on separate frames. They are much more versatile that way. If one goes down only one goes to the shop. If westbound traffic is light today I can send one west and use the other east. I can even switch with one if I need. The articulated steam loco has to be on one frame because that is the only way a steam loco can be "MU'd" and operated with one engineer and one fireman. This restriction does not apply to DEs so why stipulate it arbitrarily. If we are going to make arbitrary restrictions then I can compare the largest steam loco to the largest DE that can be operated by one crew. DEs win that one, a 12 unit lash up is going to beat any tea kettle of any size.

If we are going to apply arbitrary restrictions then why not compare locos with equal number of cylinders? The steam loco is always going to beat the DE in that comparison. And the comparison is totally meaningless. My restriction of them both being the same HP and both the same engine weight is equally arbitrary. The only thing that really matters and the only thing that can really be compared meaningfully is how much does each form of propulsion cost the RR. The bottom line is all that matters. Not HP, not tractive effort, not numbers of crews, not the cost of fuel, not the purchase price of the loco, not which power can get the tonnage across the division fastest. What matters is the cost of all these things together. Only the total package concerns the RR. The total package is quite complex.

### The Bottom Line

What is the cost of fuel for steam vs. DE? It is not just how much is used by each. It is not just the price of oil vs. coal. It is also the cost of handling that fuel, transporting it to where it is needed. The fuel handling infrastructure and manpower. You must consider the manpower it takes to run, service, and maintain each type. You must consider the availability, reliability, of each type. And you must consider the way in which the RR is operated. If steam happened to be better than DEs at high speed light weight trains that is not necessarily a plus if the RR does not operate that way. You may say that a RR SHOULD operate that way. Well it ain't your RR. Besides not all operations lend themselves to high speed light weight trains. Applying that requirement to a coal or grain haulier where he must use a LOT of HP on many small trains and you will price his service and thus the price of the commodity right through the roof. The whole package has to be compared and the total cost of each package is the only thing that can be compared. Everything else is academic nonsense.